TREATISE 


Calculus  of  Variations, 


ARRANGED  WITH  THE  PURPOSE  OF  INTRODUCING,  AS  WELL  AS  ILLUS- 
TRATING, ITS   PRINCIPLES  TO    THE   READER   BY    MEANS   OF 
PROBLEMS,    AND    DESIGNED    TO    PRESENT    IN    ALL 
IMPORTANT    PARTICULARS    A    COMPLETE 
VIEW    OF    THE    PRESENT   STATE 
OF   THE   SCIENCE. 


BY 

LEWIS    BUFFETT    CARLL,   A.M. 


NEW   YORK: 

JOHN    WILEY    AND    SONS, 

53  East  Tenth  Street, 

Second  door  west  ot  Broadway. 
1890. 


Q^ 


""15 


Copyright,  i88i,  by 
LEWIS    BUFFETT   CARLL,  A 


PREFACE. 


Thirty  years  have  now  elapsed  since  the  appearance  of 
the  treatise  on  the  Calculus  of  Variations  by  Prof.  Jellett, 
which,  although  it  had  been  preceded  by  the  smaller  work 
of  Woodhouse  in  1810,  and  of  Abbatt  in  1837,  is  justly  deemed 
the  only  complete  treatise  which  has  ever  appeared  in  Eng- 
lish. But  all  the  works  named  have  long  since  been  out  of 
print,  and  are  now  so  rare  as  not  to  be  found  in  the  majority 
of  the  college  libraries  of  the  United  States.  Moreover,  even 
Prof.  Jellett's  treatise  can  no  longer  be  regarded  as  complete, 
since  its  author  had  not  read  the  memoirs  of  Sarrus  and 
Cauchy  relative  to  multiple  integrals,  while  the  contributions 
of  Hesse,  Moigno  and  Lindelof,  and  Todhunter  were  subse- 
quent to  the  publication  of  his  work.  It  should  be  added, 
also,  that  all  the  memoirs  and  contributions  just  named  are 
contained  in  works  which  are  likewise  out  of  print,  and  are 
now  almost  as  difificult  of  access  to  the  general  reader  as  is 
that  of  Prof.  Jellett. 

These  considerations  first  led  the  author  to  undertake  the 
preparation  of  the  present  treatise,  in  which  he  has  endeav- 
ored to  present,  in  as  simple  a  manner  as  he  could  command, 
everything  of  importance  which  is  at  present  known  concern- 
ing this  abstruse  department  of  analysis. 

In  the  execution  of  this  design  the  following  method  has, 
so  far  as  possible,  been  pursued :  When  a  new  principle  is  to 
be  introduced  for  the  first  time,  a  simple  problem  involving  it 
is  first  proposed,  and  the  principle  is  established  when  re- 


IV  PREFACE. 

quired  in  the  discussion  of  this  problem.  This  having  been 
followed  by  other  problems  of  the  same  class,  the  general 
theory  of  the  subject  is  finally  given  and  illustrated  by  one  or 
two  of  the  most  difficult  problems  obtainable;  after  which 
another  principle  is  introduced  in  like  manner. 

Although  the  view  taken  of  a  variation  is  that  of  Profs. 
Airy  and  Todhunter,  and  the  methods  of  varying  functions 
are  those  of  Jellett  and  Strauch,  still  all  the  other  leading 
conceptions  and  methods  have,  it  is  hoped,  been  explained 
with  sufficient  fulness  to  enable  the  reader  to  follow  them 
when  they  occur  in  other  works. 

The  history  of  the  subject  is  also  briefly  given  in  the  last 
chapter,  it  being  believed  that  the  proper  time  for  the  presen- 
tation of  the  history  of  any  science  is  after  the  reader  has 
become  familiar  with  its  principles,  as  it  can  then,  by  the  use 
of  some  technical  terms,  be  accomplished  more  fully  within  a 
given  space. 

To  aid  the  non-classical  reader,  the  use  of  Greek  letters 
has,  with  the  exception  of  two,  whose  use  is  now  universal, 
and  which  are  explained,  been  avoided,  except  in  references, 
or  in  such  passages  as  may  be  omitted  without  serious  loss. 
Attention  is  also  called  to  the  words  br ac hist oc krone  and 
parallelepipedon,  which  are  in  this  work  spelled  according  to 
their  derivation.  The  correct  orthography  of  the  former  had 
been  previously  adopted  by  Moigno  and  Todhunter,  and  it  is 
hoped  that  it  may  be  sufficient  to  call  the  attention  of  Greek 
scholars  to  the  latter. 

One  of  the  great  obstacles  to  the  preparation  of  the  pres- 
ent treatise  has  been  the  difficulty  of  procuring  the  author- 
ities which  it  w^as  necessary  to  consult  ;  and  the  author  would 
here  return  his  thanks  to  the  officers  of  his  Alma  Mater, 
Columbia  College  ;  to  Dr.  Noah  Porter,  the  President,  and 
Mr.  A.  Van  Name,  the  Librarian,  of  Yale  College  ;  and  to  Mr. 
Walter  M.  Ferris,  of  Bay  Ridge,  L.  I.,  for  the  extended  loan 
of  rare  works  which  could  not  be  found  in  other  libraries,  or 


PREFACE.  T 

if  found,  could  not  be  had  at  home  for  that  careful  study 
which  they  in  many  cases  demanded.  The  author  is  also 
greatly  indebted  to  Lieut.  Fred.  V.  Abbot,  U.S.iV. ;  to  M.  S. 
Wilson,  Ph.B.,  to  Prof.  P.  Winter,  and  to  the  late  A.  San- 
der, Ph.D.,  all  of  the  Flushing  Institute,  for  valuable  assist- 
ance in  the  examination  of  French  and  German  works.  But 
the  greater  part  of  the  assistance  which  the  author  has  received 
was  rendered  by  his  youngest  brother,  who,  in  addition  to 
aiding  in  the  examination  of  many  works,  recopied  the  manu- 
script for  the  printer,  and  subsequently  undertook,  in  con- 
junction with  the  author  himself,  the  proof-reading  of  the 
entire  publication. 

It  having  been  found  necessary  to  publish  the  present  treatise 
by  subscription,  the  author,  supported  by  President  Barnard 
of  Columbia  College,  Prof.  J.  H.  Van  Amringe  of  the  same, 
Joseph  W.  Harper,  Jr.,  and  others,  issued  an  appeal  to  the 
public,  which  shortly  elicited  the  following  subscriptions,  the 
copies  being  placed  at  four  dollars  each : 

Seth  Low  and  A.  A.  Low,  25  copies  each. 

D.  Appleton  &  Co.,  12  copies. 

Richard  L.  Leggett  and  John  Claflin,  10  copies  each. 

A.  S.  Barnes  &  Co.,  6  copies. 

Joseph  W.  Harper,  Jr.,  Chas.  Scribner's  Sons,  Ivison,  Blake- 
man,  Taylor  &  Co.,  F.  A.  P.  Barnard,  LL.D.,  Prof.  J.  H.  Van 
Amringe,  Columbia  College  Libraries,  Gen.  Alexander  S. 
Webb,  John  H.  Ireland,  Malcolm  Graham,  Franklin  B.  Lord, 
Francis  A.  Stout,  Fred.  A.  Schermerhorn,  Frank  D.  Sturges, 
Robert  Shepard,  Edward  Mitchell,  E.  H.  Nichols,  Prof.  Felix 
Adler,  W.  Bayard  Cutting,  Hon.  Benj.  W.  Downing,  A. 
Ernest  Vanderpoel,  John  Cropper,  Willard  Bartlett,  Clarence 
R.  Conger,  Wm.  Macnevan  Purdy  and  Chas.  Pratt,  5  copies 
each. 

Prof.  C.  W.  Jones,  4  copies. 

Wm.  C.  Schermerhorn,  J.  Harsen  Rhoades,  Prof.  E.  M. 
Bass,  Henry  C.  Sturges  and  Dr.  Edw.  L.  Beadle,  3  copies  each. 


VI  PREFACE. 

Hon.  Abram  S.  Hewitt,  Gerard  Beekman,  Geo.  P.  Put- 
nam's Sons,  Dr.  Geo.  M.  Peabody,  Chas.  A.  Silliman,  Hon. 
Robt.  Ray  Hamilton,  Morgan  Dix,  S.T.D.,  Wm.  B.  Wait, 
Mrs.  Asa  D.  Lord,  Dr.  J.  W.  S.  Arnold,  Dr.  R.  W.  Witthaus, 
Rev.  Fred.  B.  Carter,  Mrs.  C.  Roberts,  R.  L.  Belknap,  Prof. 
C.  M.  Nairne,  J.  Forsythe,  D.D.,  R.  L.  Kennedy,  John  A. 
Monsell,  Robt.  Willets  and  John  F.  Carll,  2  copies  each. 

Johns  Hopkins  University,  Williams  College,  Dartmouth 
College,  National  College  of  Deaf  Mutes,  Perkins  Institute 
for  the  Blind,  Kentucky  Institute  for  the  Blind,  Indiana 
Institute  for  the  Blind,  B.  B.  Huntoon,  A.  M.  Shotwell,  Henry 
Bogert,  N.  J.  Gates,  Prof.  E.  L.  Youmans,  Prof.  Wm.  G. 
Peck,  Prof.  Henry  Drisler,  Prof.  Ogden  M.  Rood,  Prof.  Chas. 
Short,  Rev.  Spencer  S.  Roche,  W.  E.  Byerly,  Prof.  T.  H. 
Safford,  J.  P.  Paulison,  M.  M.  Backus,  A.  Wilkenson,  J.  H. 
Broully,  Geo.  H.  Mussett,  F.  L.  Nichols,  Col.  Chas.  McK. 
Loeser,  Prof.  Samuel  Hart,  Prof.  W.  W.  Beman,  S.  P.  Nash,  J. 
McL.  Nash,  O.  R.  Willis,  Ph.D.,  S.Vernon  Mann,  Hon. Wm.  H. 
Onderdonk,  Henry  Onderdonk,  Rev.  E.  A.  Dalrymple,  Gouve- 
neur  M.  Ogden,  Robt.  C.  Cornell,  Bache  McE.  Whitlock,  Geo. 
C.  Cobbe,  S.  A.  Reed,  Prof.  D.  G.  Eaton,  Dr.  D.  H.  Cochrane, 
Geo.  S.  Schofield,  Hon.  Stewart  L.  Woodford,  William  Jay, 
John  McKean,  Prof.  H.  C.  Bartlett,  Denniston  Wood,  Prof. 

A.  J.  Du  Bois,  F.  L.  Gilbert,  J.  B.  Taylor  Hatfield,  Foster  C. 
Griffith,  Hon.  Thomas  C.  E.  Ecclesine,  Malcolm  Campbell, 
Lefferts  Strebeigh,  R.  H.  Buehrle,  Prof.  H.  A.  Newton,  Wm. 
Hillhouse,  M.D.,  Jas.  L.  Onderdonk,  Wm.  B.  Patterson,  Prof. 
J.  E.  Kershner,  Francis  M.  Eagle,  Warren  Bigler,  C.  J.  H. 
Woodbury,  Rev.  Peter  J.  Desmedt,  D.  H.  Harsha,  Prof.  J. 
W.  Nicholson,  Prof.  Peter  S.  Michie,  Lieut.  S.  W.  Roessler, 
E.  F.  MiUiken,  Wm.  P.  Humbert,  Chas.  E.  Emery,  Prof.  A. 

B.  Nelson,  Adam  McClelland,  D.D.,  Prof.  H.  T.  Eddy,  Miss 
H.  L.  Baquet,  and  others  not  wishing  their  names  published, 
I  copy  each. 

Warned  by  the  experience  of  others,  the  author  was  con- 


PREFACE.  •.  Vll 

vinced  from  the  first  that  he  could  hope  to  derive  no  pecu- 
niary profit  from  a  work  like  the  present.  But  if  it  is  now 
possible  that  there  may  accrue  to  him  some  snlall  financial 
return,  this  possibility  is  due  to  the  liberality  of  his  publishers, 
who,  although  consulted  late,  and  knowing  the  unremuner- 
ative  character  of  the  work  offered,  proposed  voluntarily  to 
undertake  its  publication  upon  terms  more  favorable  than 
those  which  he  had  been  endeavoring  to  secure. 

The  acknowledgments  of  the  author  are  due  also  to  his 
printer,  S.  W.  Green's  Son,  for  the  general  excellency  of  the 
proof  furnished,  and  especially  for  his  uniform  readiness  to 
do,  without  regard  to  trouble,  whatever  was  indicated  as 
tending  to  render  the  work  more  correct  in  minor  points. 

But  while  the  author  has,  in  the  particulars  mentioned, 
received  much  assistance  from  friends,  to  whom  he  would 
return  his  unfeigned  thanks,  he  deems  it  but  just  to  himself 
to  say  that  he  has  never  enjoyed  the  acquaintance  of  any  one 
who  had  made  the  Calculus  of  Variations  the  subject  of 
extensive  study,  and  has  consequently  been  obliged  to  depend 
solely  upon  his  own  judgment  and  the  various  works  which 
he  has  consulted. 

It  is  not  therefore  believed  that  the  present  treatise  can 
be  entirely  free  from  mathematical  errors  ;  and  hence  the 
author  would  respectfully  ask  his  readers,  and  especially  those 
among  them  who  may  have  given  previous  attention  to  this 
subject,  to  indicate  any  points  in  which  his  methods  or  results 
appear  erroneous,  or  any  places  in  which  misprints  may  have 
been  allowed  to  pass  unnoticed. 

L.  B.  CARLL. 

Flushing,  Queens  Co.,  N,  Y.,  July  8,  1881. 


CONTENTS 


CHAPTER   I. 


MAXIMA  AND   MINIMA   OF   SINGLE   INTEGRALS   INVOLVING  ONE 
DEPENDENT  VARIABLE. 

Section  I. 

Case  in  which  the  Limiting  Values  of  x,  y,y\  etc.,  are  given. 

Prob.  I.  To  find  the  shortest  plane  curve  between  two  fixed  points  ;  Mode  of 
comparing  curves,  page  i.  Meaning  of  the  term  variation,  and  method  of  de- 
noting it,  2.  No  variation  of  the  independent  variable  necessary  ;  Curve  varied 
by  means  of  its  tangents  only,  3.  Development  of  81,  or  the  difference  of  length 
of  two  consecutive  curves,  4.  Review  of  the  theory  of  maxima  and  minima  in 
the  differential  calculus  ;  A  maximum  not  always  the  greatest,  nor  a  minimum  the 
least,  value  of  a  function,  5,     Extension   of  this  theory  to  present  problem,   7. 

Faulty  solution  obtained,  8.     6>(«)  =  -~,  9.     True  solution  obtained,  10.     Why 

first  method  of  solution  failed,  11.  Term  of  the  second  order  examined  ;  Coeffi- 
cient of  dy"^dx  must  be  of  invariable  sign,  12.  No  maximum  nor  minimum  when 
it  vanishes  permanently;  Previous  solution  not  absolute,  13.  Power  of  the  new 
method;  The  constants  a  and  b  determined,  14.  Prob.  II.  To  find  the  plane 
brachistochrone,  or  curve  of  quickest  descent  of  a  particle,  between  two  fixed 
points,  15.  Simplification  by  omitting  constant  factor;  General  solution,  16. 
Particular  solution,  i8„  Term  of  the  second  order  positive;  Limitation  of  solu- 
tion, ig.  Occurrence  of  infinite  quantities,  20.  Case  2.  Same  problem  with  hori- 
zontal as  independent  variable,  21.  Ordinary  maxima  or  minima  of  functions 
involving  several  variables,  22.  Extension  to  present  problem,  23.  dy  and  dy 
not  independent,  24.  Terms  of  the  second  order  examined,  26.  Limitation  of  the 
solution,  28.  Derived  curve  and  primitive  curve,  29.  Prob.  III.  To  find  the 
curve  which,  with  its  evolute  and  extreme  radii  of  curvature,  shall  enclose  a  min- 
imum area,  30.     Terms  of  the  second  order,  33.     Limitation  of  solution,  35. 


X  CONTENTS. 

Section  II. 

Case  in  which  the  Litniting  Values  of  x  only  are  given. 

General  discussion,  35.  Mode  of  transforming  dUhy  integration,  37.  Final 
form  of  dU,  39.  The  coefficients  of  (5>o,  <5>i,  8ya' ,  8yi  ,  etc.,  must  severally  van- 
ish, 40.     The  coefficient  of  dydx  in  the  integral  must  also  vanish,  41.         Prob. 


IV.  To  maximize  or  minimize  the  expression     /     f  {y")  dx,  42.         Prob.  V.  To 

nx-i 
limize     /      (/'^  —  2y)   dx,  44.  Probs.   I.,   II.    and   III.   re- 


'Xo 

maximize  or  minimize 

'x^ 

sumed,  48.  Number  and  determination  of  the  constants  in  the  complete  integral 
oi  M :=  o,  52.  Three  exceptional  cases,  55.  Integrability  of  M=o;  Formulae 
(A),  (B),  (C)  and  (D),  61.  Prob.  VI.  To  find  the  solid  of  revolution  which  will 
experience  a  minimum  resistance  in  passing  through  a  fluid  in  the  direction  of  the 
axis  of  revolution,  66.  Prob.  VII.  To  find  the  surface  of  revolution  of  mini- 
mum area,  70.         Prob.  VIII.  To  find  formula  for  terms  of  second  order  when 


pxi 
U-=    I       yf{y')dx,  77.     Application  to  Prob.  VI.,  80. 

VfXo 


Section  III. 

Case  in  which  the  Limiting  Values  of  x  also  are  variable. 

Prob.  IX.  To  find  the  curve  of  minimum  length  between  two  curves,  all  in  the 
same  plane;  Mode  of  varying  the  required  curve  when  x^  and  xi  are  variable,  81. 
General  solution  must  be  independent  of  conditions  at  the  limits,  84.  8y^,  dx^, 
6yi  and  dxi  cannot  be  independent,  85.  Formulae  for  eliminating  dyo  and  dvi,  86. 
Application  to  present  problem,  87.  Terms  of  the  second  order  considered,  88. 
Prob.  X.  To  find  the  plane  brachistochrone,  the  particle  starting  from  any  hori- 
zontal, 90.  General  theory  of  the  subject,  93.  Prob.  XI.  The  evolute  problem 
with  limiting  curves,  94.  Prob.  XII.  Minimum  surface  of  revolution  with  varia- 
ble limits,  95. 

Section  IV. 

Case  in  which  some  of  the  Limiting  Values  of  x,  y,  y',  etc.,  enter  the 
General  Form  of  V. 

Prob.  XIII.  To  find  the  brachistochrone  between  two  curves,  the  particle  start- 
ing from  the  upper  curve,  96.  Terms  of  the  second  order,  loi.  General  theory 
of  the  subject,  104. 


CONTENTS.  XI 


Section  V. 


Case  in  which  U  is  a  Mixed  Expression;    that  is,  contains  an  Integral 
together  with  Terms  free  from  the  Integral  Sign. 

nx-^      y" 
Prob.  XIV.  To  maximize  or  minimize    /     y")  —  dx,  105.     General  theory  of 

the  subject,  107.  ^"^^         ^ 

Section  VI. 
)       Relative  Maxima  and  Minima. 

Prob.  XV.  To  find  the  plane  curve  of  given  length  which,  with  its  extreme 
ordinates  and  the  axis  of  x,  shall  contain  a  maximum  area;  Restriction  of  derived 
curves  and  use  of  the  terms  absolute,  relative  and  isoperimetrical,  108.  Why- 
ordinary  method  of  solution  fails,  109.  Special  condition  established,  no.  Solu- 
tion effected,  in.  Determination  of  the  constants,  113.  Bertrand's  proof  of 
Euler's  method,  114.  Immaterial  whether  we  regard  the  curve  or  area  as  con- 
stant, 116.  Terms  of  the  second  order  considered,  117.  Problem  considered 
when  limits  are  variable,  iig.  Prob.  XVI.  To  find  the  solid  of  revolution  of 
given  surface  and  maximum  volume,  the  generatrix  meeting  the  axis  of  revolu- 
tion, at  two  fixed  points,  122.  Constants  determined;  First  notice  of  discontinuity, 
124.  Condition  when  limits  are  variable,  126.  Character  of  the  generatrix  when 
unrestricted,  131.  Prob.  XVII.  To  find  the  curve  which  a  uniform  cord  must 
assume  when  its  centre  of  gravity  is  as  low  as  possible,  133.  Prob.  XVIII. 
Case  of  James  Bernoulli's  problem,  138.  Prob.  XIX.  To  find  the  solid  of  revo- 
lution of  given  mass  whose  attraction  upon  a  particle  situated  upon  the  axis  of 
revolution  shall  be  a  maximum,  141.  Prob.  XX.  To  find,  as  in  Prob.  VI.,  the 
solid  of  minimum  resistance,  supposing  its  mass  and  base  to  be  given,  144. 
Prob.  XXI.  To  find  the  solid  of  revolution  of  given  mass  and  minimum  moment 
of  inertia  with  respect  to  an  axis  perpendicular  to  that  of  revolution  through  its 
middle  point,  148. 

Section  VII. 

Case  in  which  V  is  a  Function  of  Polar  Co-ordinates  and  their  Differential 

Coefficients. 

Prob.  XXII.  To  find  the  path  of  minimum  action  for  a  revolving  particle, 
attracted  toward  a  fixed  point,  according  to  the  Newtonian  law,  151.  Prob. 
XXIII.  To  find  the  plane  closed  curve  of  given  length  which  will  enclose  a  maxi- 
mum area,  158. 


XU  •  CONTENTS. 

Section  VIII. 

.  Discri7nmation  of  Maxima  and  Minima  {Jacobi's  Theorem). 

Case  I.  V  supposed  to  be  a  function  of  x,  y  and y'  only,  163.  Rules  for  apply- 
ing theorem  in  this  case,  174.  Prob.  XXIV.  To  apply  theorem  of  Prob.  I.,  175. 
Prob.  XXV.  Application  to  the  first  case  of  Prob.   II.,   176.  Prob.   XXVI. 

Application  to  Prob,  XXII.,  involving  polar  co-ordinates,  178.  Geometrical  illus- 
tration of  theorem  for  this  case,  182.         Prob.   XXVII.     To  apply  theorem  to 

I      y^f{y)dx,   184,      Discussion    of   Prob.  VIII.    completed,    188.  Prob. 

XXVIII.  To  apply  theorem  to  Prob.  VII.,  189.  Probs.  XXIX.  and  XXX. 
To  apply  theorem  to  the  second  case  of  Prob.  II.,  and  also  to  Prob.  XVI.,  193. 
General  discrimination  in  cases  of  relative  maxima  and  minima  ;  Euler's  method 
and  Jacobi's  theorem  defective,  197.  Two  exceptions  to  Case  i,  202.  General 
view  of  Jacobi's  method,  204.  Two  lemmas,  206.  Case  2.  V supposed  to  be  a 
function  of  x,  y,  y'  and y"  only,  20g.  Prob.  XXXI.  To  apply  theorem  to  Prob. 
v.,  226.  General  rules  for  applying  theorem  in  this  case,  230.  Three  excep- 
tions to  Case  2,  231.  Case  3.  V  supposed  to  be  a  function  of  x,y,y',  .... 
I'W,  232,  Four  exceptional  forms  of  F,  241.  Connection  of  the  variations  of 
U  when  we  take  successively  x  and  y  as  the  independent  variable,  242. 

Section  IX. 

Discontinuous  Solutions. 

Prob.  XXXII.  Maximum  solid  of  revolution  resumed,  248.  Failure  of  former 
methods,  and  discontinuous  solution  proposed,  250.  Sign  of  dy,  8y\  etc.,  not 
always  in  our  power;  M  not  always  zero;  No  appeal  to  terms  of  second 
order  necessary,  251.  Discontinuous  solution  confirmed  by  foregoing  principles, 
253.  Cause  of  discontinuity;  Explicit  and  implicit  conditions,  255.  Prob. 
XXXIII.  -Prob.  V.  resumed  with  conditions,  256.  General  theory  of  discontin- 
uous solutions,  264.  Prob,  XXXIV.  To  solve  Prob.  VII.  when  a  catenary  is 
impossible;  General  discussion,  267.  Two  solutions  admissible,  and  formulae 
for  determining  which  will  give  the  smaller  surface,  273.  ProTa.  XXXV.  To 
determine  the  path  of  least  or  minimum  action  for  a  projectile  acted  upon  by 
gravity  only,  275.  Jacobi's  theorem  applied  to  problem,  279.  Discontinuous 
solutions  considered,  281.  Formulae  when  two  solutions  are  admissible,  283. 
Prob.  XXXVI.  To  seek  a  discontinuous  solution  for  Prob.  XXII,  involving  polar 
co-ordinates,  285.         Prob.  XXXVII.  To  determine  under  certain  conditions  the 


CONTENTS.  Xlll 

quickest  or  brachistochrone  course  of  a  steamer  between  two  ports  on  a  stream,  287. 
Prob.  XXXVIII.  The  brachistochrone  course  of  a  vessel  when  length  of  course  is 

fixed,  291.         Prob.  XXXIX.  To  maximize  or  minimize    /       \  ay"-^-\-—  l  dx, 

299.  Prob.  XL.  To  determine  the  discontinuous  solution  in  Prob.  XV.,  302. 
Prob.  XLI.  To  determine  the  discontinuous  solution  in  Prob.  XIX.,  306. 
Prob.  XLII.  To  find  the  discontinuous  solution  in  Prob.  XXL,  309.  Prob. 
XLIII.  To  find  the  discontinuous  solution  in  Prob.  XX.,  311.  Prob.  XLIV.  To 
find  the  plane  brachistochrone  between  two  points,  the  particle  not  to  pass  with- 
out a  certain  circular  arc,  not  exceeding  a  quadrant,  322. 


Section  X. 

Other  Methods  of  Variations. 

First  Method.  Variations  ascribed  to  x  also  throughout  U\  Method  illustrated 
geometrically,  328.  General  expression  for  8  U,  329.  Expressions  for  dy' ,  8y\ 
etc.,  330.  (5  i/ reduced  and  compared  with  the  usual  form,  331.  General  solu- 
tions, and  also  terms  at  the  limits,  the  same  as  by  previous  method,  332.  Merits 
of   the   two  methods,    334.  Second  Method.     The  arc   ^   taken  as   the   inde- 

pendent variable,    but   not  varied.         Prob.  XLV.    To  maximize  or  minimize 

/    ^  f  {x>  y)Vi-\-y^dx;  J  made  the  independent  variable,  335.     Method  illus- 

trated  ;  s  must  be  varied  at  the  limits,  336.  56^  obtained  ;  dx  and  dy  not  inde- 
pendent, 337.  Method  of  Lagrange  introduced,  338.  Expressions  for  dx',  dx", 
etc.,  dy',  dy",  etc.,  339.  Formulae  for  dxi  and  dyi  when  limits  are  fixed,  341. 
Same  when  limits  are  variable,  342.  General  formula  of  solution,  344.  Applica- 
tions ;  Final  limiting  equations  the  same  as  when  x  is  the  independent  variable, 

345.         Prob.    XLVI.  To   maximize  or  minimize     /        (v  Vi  -^  y'^  -{-  n)   dx  = 
(v  -|-  Mx')  ds,  V  and  u  being  functions  of  x  and  y  only,  346.     General  for- 


t/so 


'So 
mula  of  solution,  348.     Applications,   349.         Prob.   XLVI  I.    To    maximize  or 

minimize    /       Vds,   V  being  a  function  of  r,  the  radius  of  curvature  of  a  plane 


t/so 


curve,  350.  Application  to  Prob.  III.  and  to  the  problem  of  the  elastic  spring, 
353.  Third  Method.  Vzx'idilions  ascribed  to  the  independent  variable  j-;  Method 
illustrated,  354.  General  form  oidU,  355.  Expressions  for  dx',dx",  dy' ,  dy", 
etc.,  356.  dx  and  dy  not  independent,  and  method  of  Lagrange  necessary,  357. 
Method  applied  to  Probs.  XLV.,  XLVI.  and  XLVIL,  358. 


XIV  CONTENTS, 


CHAPTER   II. 


MAXIMA  AND   MINIMA  OF  SINGLE   INTEGRALS   INVOLVING  TWO 
OR    MORE    DEPENDENT    VARIABLES. 


Section  I. 

Case  in  which  the  Variations  are  unconnected  by  any  Equations. 

Prob.  XLVIII.  To  find  the  shortest  curve  in  space  between  two  points,  curves 
or  surfaces;  Former  principles  extended,  363.  Expressions  for  5y">,  52<»),  <5w("), 
etc.,  364.  Constants  determined,  366.  Terms  of  the  second  order  examined,  367. 
x^  and  xx  variable,  368.  Prob.  XLIX.  To  find  the  brachistochrone  in  space  for 
a  particle  descending  from  one  fixed  point,  curve  or  surface  to  another,  372. 
Number  and  determination  of  the  constants  in  the  general  solution  of  problems  of 
this  class,  374.         Prob.  L.  To  establish  the  principle  of  least  action,  375. 


Section  II. 

Case  in  which  the  Variations  are  connected  by  Equations,  Differential  or 

other. 

Prob.  LI,  To  find  the  shortest  or  geodesic  line  upon  the  surface  of  a  sphere, 
379.  General  discussion  of  the  method  of  Lagrange,  384.  Number  of  the  con- 
stants and  ancillary  equations,  385.     Lagrange's  method  extended,  386.         Prob. 

LIL  To  maximize  or  minimize     /       v  V 1 -\- y"^ -\- z'^  dx,  v  being  a  function  of 

Jxq 
X,  y  and  z,  the  connecting  equation  being /(.Jtr,  y,  z)  =0  ,  388.  Prob.  LIII.  To 
find  the  shortest  or  geodesic  curve  upon  any  surface,  396.  Prob.  LIV.  To 
find  the  geodesic  curve  upon  the  surface  of  a  spheroid,  399.  Prob.  LV.  Tp 
find  the  brachistochrone  upon  a  given  surface,  the  particle  being  urged  by  any 
system  of  forces,  401.  Prob.  LVI.  To  find  the  curve  of  minimum  length  be- 
tween two  points  in  space,  the  radius  of  curvature  being  a  constant,  405.  Dis- 
continuous solution,  410.     Miscellaneous  observations,  418. 


CONTENTS.  XV 

CHAPTER   III. 

MAXIMA   AND    MINIMA   OF   MULTIPLE    INTEGRALS. 


Section  I. 

Case  in  which  U  is  a  Double  Integral,  the  Limiting  Values  of  x,  y,  z,  etc. , 

being  fixed. 

Prob.  LVII.  To  find  the  surface  of  minimum  area  terminating  in  all  directions 
in  a  given  closed  linear  boundary;  Mode  of  comparing  surfaces,  422.  View  of 
double  integration,  423.  Projected  contour,  and  sign  of  substitution,  424.  Varia- 
tions introduced,  427.  Principles  of  maxima  and  minima  extended;  Formulae 
for  8p,  dq,  dr,  8t,  etc.,  429.  Solution  effected,  430.  Terms  of  the  second  order; 
Limitation  of  solution,  434.  Form  of  M  and  nature  of  its  integral  when  F  is  a 
function  of  x,  y,  z,  p  and  q  only,  435.  Prob.  LVIII.  The  intersept  problem, 
437. 

Section  II. 

Forjnulce  necessary  for  the  Transforinatio7i  of  the  Variations  of  a 

Mutiple  Ifitegral.  439 


,    Section  III. 

Maxima  and  Minima  of  Double  Integrals  with   Variable  Limits 

Prob.  LIX.  To  discuss  Probs.  LVII.  and  LVIII.  when  the  limiting  values  of  x 
and  y  only  are  fixed  ;  New  form  of  8  U,  and  terms  at  the  limits  explained,  447. 
Conditions  when  ^is  to  be  a  maximum  or  minimum,  450.  Results  applied  to 
problem,  452.  Prob.  LX.  To  discuss  Probs.  LVII.  and  LVIII.  when  the 
required  surface  is  to  touch  one  or  m.ore  given  surfaces;  that  is,  when  the  limit- 
ing values  of  x  and  y  are  also  variable  ;  x  and  y  varied  at  the  limits  only,  454. 
Mode  of  obtaining  5  f/ to  the  second  order,  455.  Terms  of  the  first  order  only  con- 
sidered, 459.  Some  limiting  surface  necessary  for  a  maximum  or  a  minimum; 
Equations  between  dz,  Dy  and  Dx  at  the  limits,  461.  Final  form  of  terms 
at  the  limits,  463.  Application  to  present  problem,  464.  Mode  of  determin- 
ing  the   arbitrary  functions  involved   in    the   complete  integral  of  M  =0,    465. 


XVI  CONTENTS. 

Occurrence  of  infinite  quantities,  469.         Prob.  LXI.   To  maximize  or  minimize 
/        /         ^p'^-\-(f  dy  dx,    470.  Prob.    LXII.     To   maximize   or  minimize 

I     ^  I    ^  {z  —  px  —  qy)dy  dx,  while     I        I         Vf^-[-q^dydx   remains    con- 

stant;  Mode  of  extending  Euler's  method,  475.  Terms  of  the  second  order,  480. 
Prob.  LXI  1 1.  To  find  the  surface  of  lowest  centre  of  gravity,  481.  Prob.  LXIV. 
To  find  the  minimum  surface  covering  a  given  volume  upon  a  plane,  483.  Prob. 
LXV.  To  discuss  the  case  when  K  is  a  function  of  x,  y,  z,  p,  q,  r,  s  and  /,  485. 
Imperfection  of  preceding  theory,  490.     Exceptional  forms  of  V,  492. 


Section  IV. 

Extension  of  Jacobi's    Theorem   to  the   Discrzinination   of  Maxima   and 
Minima  of  Double  Integrals.  493 


Section  V. 
Maxima  and  Minima  of  Triple  Integrals. 

Prob.  LXVI.  To  minimize  j-^^  H'  H^  ^i  +/-'  +q^-{-  ^dzdydx,  u  being 
e/xo  e/2/0  «^^o 
the  density  of  a  body  of  given  form,  position  and  mass  ;  Triple  integration  and 
limiting  faces  explained,  498.  Mode  of  finding  ^  U  when  the  limits  of  x,  y  and 
z  are  fixed,  500.  Conditions  when  ^  is  to  be  a  maximum  or  a  minimum,  503. 
Problem  solved,  505.  Form  of  (J  f/ when  the  limits  of  ;r,  _j'  and  z  are  variable, 
506.  Conditions  when  U  is  to  be  a  maximum  or  a  minimum,  509.  du,  Dz,  Dy 
and  Z>.r  not  independent ;  Equations  between  them,  510.  Conditions  resumed, 
5". 


Section  VI. 

Another  View  of  Variations. 

General  explanation,   514.         Prob.  LXVI  I.  To  find  the  solid  of  maximum 
volume,  524. 


CONTENTS,  XVU 


CHAPTER   IV. 

APPLICATION   OF  THE   CALCULUS    OF    VARIATIONS   TO   DETER- 
MINING THE   CONDITIONS   WHICH    WILL    RENDER    A 
FUNCTION   INTEGRABLE   ONE   OR   MORE   TIMES. 

Section  I. 
Case  in  which  there  is  but  One  Independent  Variable. 

Prob.  LXVIII.  To  maximize  or  minimize    /       -<  " =V-  +  ^^  >■  dx,  533. 

M  found  to  vanish  identically,  and  U  to  be  integrable  ;  General  theory  deduced 
for  similar  cases,  534.  Prob.  LXIX.  To  determine  the  conditions  which  will 
XQndQT  f  {x,  y,  y' ,  .  .  .  .  y^))  integrable  more  than  once,  536.  Prob.  LXX.  To 
determine  the  conditions  which  will  render /(x,j,y,  .  .  .  .  }^'^^,  z,  z' ,  .  .  .  .  s('»))^x 
immediately  integrable,  537. 


Section  II. 
Case  in  which  there  are  Two  Independe^it  Variables. 

Prob.  LXXI.  To  determine  the  conditions  which  will  render    /     I  f  {x,  y,  z, 
p,  q)  dydx  reducible  to  a  single  integral,  538. 


CHAPTER  V. 

HISTORICAL    SKETCH    OF    THE    RISE    AND    PROGRESS    OF  THE 
CALCULUS  OF  VARIATIONS.  54I 


NOTES. 


Note  to  Lemma  I.,  557.  Note  to  Lemma  II.,  560. 

Note  to  Art.  369,  562.  Note  to  Art.  372,  568. 


CALCULUS  OF  VARIATIONS. 


CHAPTER  I 


MAXIMA  AND   MINIMA  OF  SINGLE   INTEGRALS   INVOLVING  ONE 
DEPENDENT   VARIABLE. 


Section   I. 

CASE  IN  WHICH  THE  LIMITING  VALUES  OF  X,  F,   F',  ETC.,  ARE 

GIVEN 

Problem  I. 

I.  Suppose  it  were  required  to  find  the  shortest  plane  curve  or 
line  which  can  be  drawn  between  tzvo  fixed  points. 

Let  A  CB  be  the  required  line,  which  is  of  course  straight, 
and  AEB  any  other   line   derived    from  the  first  by  giving 


X  ox 

indefinitely  small  increments  to  any  or  all  of  its  ordinates, 
while  the  corresponding  values  of  x  remain  unaltered.  Then 
the  line  ACE  must  be  shorter  than  the  line  AEB. 

This  remark  would  be  equally  true  if  the  changes  in  the 


2  CALCULUS  OF    VARIATIONS. 

ordinates  oi  AB  had  not  been  made  indefinitely  small ;  but 
then,  even  if  the  second  line  were  shown  to  be  longer  than  the 
first,  we  could  not  be  certain  that  some  third  line,  lying  a  little 
nearer  the  first,  might  not  be  shorter  than  either.  Thus  it 
will  be  seen  that  questions  may  arise  which  require  an  investi- 
gation of  that  increment  which  a  curve  would  receive,  not 
from  any  change  in  the  values  of  x,  nor  in  the  values  of  the 
co-ordinates  of  the  fixed  extremities,  but  from  indefinitely 
small  changes  in  the  values  of  y  throughout  the  whole  or  a 
portion  of  the  curve ;  thus  altering  in  a  slight  degree  the 
functional  relation  which  previously  subsisted  between  x 
and  y. 

2.  Now  the  general  expression  for  the  length  of  any  plane 
curve  between  two  fixed  points  is 

^=/V^p"+^^  .  (I) 

in  which  the  suffix  i  relates  to  the  upper,  and  o  to  the  lower 
limit  of  integration,  and  this  expression  cannot  be  integrated 
so  long  as  y  is  an  unknown  function  of  x. 

Hence,  in  determining  the  increment  which  will  result 
to  a  curve  from  an  indefinitely  small  change  in  its  form,  we 
shall  be  concerned  with  two  species  of  small  quantities :  first, 
those  changes  which  x  and  y  undergo  as  we  pass  from  one 
point  to  another  indefinitely  near  or  adjacent  on  the  same  curve, 
which  are  denoted  by  dx  and  dy,  these  being  necessary  for 
the  general  expression  of  /  in  (i) ;  and  secondly,  that  change 
which  y  undergoes  as  we  pass  from  a  point  on  one  curve  to 
a  point  on  another  curve  indefinitely  near  or  adjacent,  the 
value  of  X  being  unaltered.  These  latter  quantities  are  called 
variations,  and  are  denoted  by  the  Greek  letter  d,  delta,  or  d. 

Thus  dy  is  read,  the  variation  of  y  ;  -j— ,  the  variation  of  -7-,  etc. 

As  another  illustration  of  the  difference  between  these  two 


SHORTEST  PLANE    CURVE  BETWEEN    TWO  POINTS.        3 

classes  of  quantities,  we  might  say  that  dy  as  used  in  (i)  is 
the  difference  between  two  consecutive  states  of  the  same 
function  of  x,  while  Sy  is  the  difference  between  two  consecu- 
tive or  adjacent  functions  taken  for  the  same  value  of  x.  The 
use  of  this  symbol  S  is  due  to  Lagrange,  and  while  it  prevents 
confusion,  it  also  suggests  the  character  of  the  variation  as  a 
species  of  differential.  It  is  plain  that  we  can  vary  the  form 
of  a  curve  which  terminates  in  two  fixed  points  in  any  man- 
ner we  please,  by  simply  giving  suitable  changes  to  its  ordi- 
nates  without  varying  its  abscissas,  and  we  shall  therefore  at 
present  ascribe  no  variation  to  the  independent  variable  x, 
but  simply  to  the  dependent  y  or  to  its  differential  coefficients 
wnth  respect  to  x, 

3.  Resuming  equation  (i),  we  will  now  show  how  to  find 

SI,  or  that  increment  which  /  would  receive,  not  from  any 

change  in  the  limits  of  integration,  but  from  an  inappreciably 

small  alteration  in  the  value  of  jr  as  a  function  of  x.     We.  shall 

dy  d^v 

in  general  put  y'  for  -p,  y"  for  ^i-^,  etc.     Then  we  have 

dx""  -k  dv" 
dx^  +  df  =  — ^F^  d^  =  (I  +y^)  dx^; 

hence  (i)  becomes 

^=ff'y^i~+rdx.  (2) 

0  1 

It  will  be  seen  that  y  does  not  occur  directly  or  explicitly 
in  the  last  equation ;  but  since  y'  represents  the  natural  tan- 
gent of  the  angle  which  a  tangent  to  the  curve  at  any  point 
makes  with  the  axis  of  x,  it  is  clear  that  the  form  of  this  curve 
can  be  also  altered  at  pleasure  by  giving  suitable  variations 
to  the  slopes  of  these  tangents,  and  that  if  these  variations  be 


4  CALCULUS  OF    VARIATLONS. 

indefinitely  small,  the  remarks  that  have  been  made  regarding 
dy  will  be  equally  true  regarding  dy' . 
Equation  (2)  may  be  written 

where 


Now  in  F  change  y'  into  y  +  Sy'. 

Then  the  new  state  of   V,  being  denoted  by   V\  may  be 
developed  by  the  extension  of  Taylor's  Theorem,  thus : 

where,  following  the  analogy  of  differentials,  we  write  6/% 
d/\  etc.,  for  {dyj,  (dyj,  etc.  Hence,  if  we  call  V-  V^SV, 
we  have 

in  w^hich  -^„  etc.,  are  the  partial  differential  coefficients  of  V 
with  respect  to  y . 

whence,  if  we  change  V  into  V\  dx  remaining  unaltered,  and 
denote  the  new  state  of  /  by  /',  we  shall  have 

V  =         V'dx, 

and  calling  V  —  /,  dl,  we  arrive  at  the  equation 


SHORTEST  PLANE   CURVE  BETWEEN   TWO  POINTS.        5 

4.  Before  proceeding  it  may  be  well  to  advert  to  the 
theory  of  maxima  and  minima,  as  developed  by  the  differ- 
ential calculus- 

A  function  is  said  to  be  a  'maximum  when  its  value  is 
greater,  and  a  minimum  when  its  value  is  less,  than  that 
which  it  would  have  if  any  or  all  of  its  variables  should  receive 
indefinitely  small  increments,  either  positive  or  negative. 
Thus  while  the  greatest  value  of  a  function,  if  not  infinite,  is 
always  a  maximum,  it  does  not  follow  that  every  maximum 
is  the  greatest  value  of  which  the  function  is  capable.  Neither 
is  the  greatest  value  in  every  case  the  only  maximum.  The 
foregoing  remarks  apply  equally  to  a  minimum,  it  being  only 
necessary  in  either  case  to  compare  the  supposed  maximum 
or  minimum  state  of  the  function  with  the  value  of  the  states 
which  immediately  precede  and  succeed  it. 

Taking,  for  simplicity,  a  function  of  a  single  variable,  this 
state  is  determined  and  comparison  effected  as  follows :  Let 
/  be  any  function  of  x  and  constants,  and  change  x  into  x  -J-  //. 
Then  if  we  develop  f ,  the  new  state  of  the  function,  by  Tay- 
lor's Theorem,  and  subtract  the  original  state,  we  shall  have 


f'-f=t'^^h:^'''-^-^       w 


h  being  either  positive  or  negative. 

We  shall  denote  this  series  by  5.  Then,  if  /  is  to  be  a 
maximum  or  minimum,/'— /must  be  negative  in  the  former 
case  and  positive  in  the  latter,  independently  of  the  sign  of  h. 
But  il  no  differential  coefficient  in  S  become  infinite,  and  we 
make  //  indefinitely  small,  the  sign  of  S  will  either  depend 
upon  that  of  its  first  term,  which  cannot  be  independent  of  h. 


O  CALCULUS  OF  VARLATIONS, 

or,  if  that  term  reduce  to  zero,  upon  the  sign  of  the  first  that 
does  not. 

Now  if  this  term  be  of  an  odd  order,  its  sign  would  be 
affected  by  any  change  in  that  of  h ;  but  if  of  an  even  order  it 
would  not,  since  h  must  be  real.  Hence  any  value  of  x  which 
would  render /a  maximum  or  minimum  must  at  least  satisfy 

the  equation  -3—  =  o,  and  the  roots  of  this  equation  furnish  us 

with  trial  values  of  x^  which,  when  substituted  in  the  remain- 
ing terms  of  5,  must  render  the  second  term  negative  for  a 
maximum  and  positive  for  a  minimum,  or  must  fulfil  the 
same  condition  for  some  other  term  of  an  even  order,  having 
reduced  those  which  preceded  it  to  zero ;  and  we  must  reject 
those  values  of  x  which  do  not  satisfy  these  conditions. 

It  may  also  be  useful  to  observe  that  -j-  does  not  repre- 
sent the  exact  ratio  of  the  increments  of  /  and  x,  dx  being 
infinitesimal,  but  merely  the  limit  of  that  ratio ;  that  is,  the 
value  toward  which  it  may  be  made  to  approach  to  within 
any  assignable  limit,  but  which  it  can  never  actually  equal,  it 
being  meaningless  to  say  that  dx  ever  really  becomes  zero. 

Or,  better,  we  may  regard  -j-  as  merely  a  function  derived 

from  f  by  certain  algebraic  methods  which  accord  with  the 
rules  of  differentiation ;  and  the  same  remarks  will  apply  to 
the  higher  differential  coefficients  of  /. 

Hence,  since  these  coefficients  are  entirely  Independent  of 
any  increment  which  /  actually  receives,  we  may,  without 
altering  any  of  them,  replace  h  in  (4)  by  dx^  Sx^  or  any  other 
infinitesimal  we  please. 

6.  If  the  roots  of  the  equation  -j-  —  o  comprised  all  the 

values  of  x  which  could  render  /  a  maximum  or  minimum, 
still,  since  /  might  be  capable  of  several  maxima  or  minima, 


/ 


SHORTEST  PLANE   CURVE  BETWEEN   TWO  POINTS.         J 

we  would  have  to  determine  which  maximum  would  be  the 
greatest,  or  which  minimum  the  least;  although  the  deter- 
mination would  in  general  be  easy  enough.  But  the  equa- 
tion in  question  does  not  give  all  the  required  values  of  x. 
For,  if  any  of  the  differential  coefficients  in  (4)  become  infinite, 
the  reasoning  of  the  last  article  will  no  longer  hold  true.  In 
fact,  it  is  well  known  that  /  can  become  a  maximum  or  mini- 
mum when  its  first  differential  coefficient  is  infinite,  or  when 
the  same  is  finite  while  the  second  is  infinite.  These  instances 
are  examples  of  what  are  often  termed  failing  cases  of  Tay- 
lor's Theorem — although,  strictly  speaking,  the  theorem  does 
not  fail  at  all,  only  the  development  becomes  useless  from  its 
indeterminate  character,  and  that  not  from  any  imperfection 
in  the  theorem  itself,  but  owing  to  the  existence  of  such  con- 
ditions as  to  render  impossible  an  entirely  finite  development 
of  the  form  required. 

6.  Since  the' value  of  h  in  (4)  is  altogether  independent  of 
its  coefficients,  and  might  be  replaced  by  dx,  dx,  or  any  other 
symbol  we  please,  it  is  clear  that  the  form  in  which  we  have 
expressed  d/  in  (3)  is  analogous  to  that  of  5  or/'—/,  except 
that  each  term  in  SI  is  multiplied  by  dx,  and  is  under  an 
integral  sign,  and  that  the  function  taken  is  one  of  y  and  con- 
stants, among  which  x  is  reckoned. 

Considering  the  first  term  of  that  expression,  viz., 


/ 


dy 


we  see  that  by  taking  Sy  indefinitely  small  throughout  the 
curve  we  may  ultimately  render  this  term  greater  than  the 
sum  of  the  others,  unless,  indeed,  that  integral  becomes  zero 
for  all  possible  values  of  Sy;  it  being  understood  that  the 
variation  of  any  quantity  is  to  be  always  infinitesimal  as  com- 
pared with  that  quantity.  It  is  also  clear  that  if  we  change 
the  sign  of  Sy  throughout  the  integral — that  is,  of  each  sy. 


■o  CALCULUS  OF  VARIATIONS. 

leaving  its  minute  numerical  value  unaltered — we  shall  also 
change  the  sign  of  the  above  integral,  while  the  sign  of  the 
succeeding  integral  in  (3)  will  remain  unchangedi 

7.  From  an  examination  of  the  figures,  Art.  i,  it  will  be 
seen  that  if  ACB  be  the  minimum  line  between  two  fixed 
points,  and  we  draw  a  second  in  any  manner  we  please  by 
giving  infinitesimal  variations  to  y' ,  we  may  also  draw  a  third 
line  by  giving  to  y'  variations  numerically  equal  but  of  oppo- 
site sign.  Then,  ^\\\cq  ACB  is  a  minimum,  I'  —  I  or  61  must  be 
positive  ;  /'  being  the  length  of  either  of  the  lines  ACB, 

Hence,  from  the  reasoning  of  the  last  article,  we  must  have 

since  otherwise  61  could  not  be  of  invariable  sign,  as  its  sign 
would  be  the  same  as  that  of  the  above  integral,  which  could 
be  made  to  vary  by  changing  that  of  6y' .  Moreover,  the  sec- 
ond term  in  6L  viz., 


11  rr 


must  become  positive;  or  if  it  reduce  to  zero,  some  other 
term  of  an  even  order  must  become  positive  for  all  values  of 
dy\  all  the  preceding  terms  having  reduced  to  zero. 

But,  as  in  the  differential  calculus,  the  foregoing  is  based 
upon  the  supposition  that  none  of  the  differential  coefficients 
of  V  in  (3)  become  infinite  within  the  limits  of  integration,  or, 
in  other  words,  that  V —V  is  throughout  these  limits  capa- 
ble of  a  finite  development  by  Taylor's  Theorem,  where  V 
denotes  what  V  becomes  when  we  change  y'  into  y'-\-  6y' , 

8.  We  may  now  proceed  to  a  full  solution' of  the  problem. 
We  have 


SHORTEST  PLANE   CURVE  BETWEEN  TWO  POINTS. 


V=V,+y\     ^y-    ^-^^           y. 

d'V 

I              _    I       dW  _            3/           _ 

3/ 

dy-  - 

^(i+/T    ^"  "^y^      ^i+/T 

v^ 

Hence,  as  these  and  the  succeeding  partial  differential  coeffi- 
cients of  Fwith  respect  to  y'  are  all  finite,  we  can  develop  /' 
by  Taylor's  Theorem,  and  equation  (3)  gives 

in  which  we  have  first  to  consider  the  expression 

£/-sydx  =  o.  (6) 

v' 
This  equation  is  of  course  satisfied  by  making  ~  zero, 

which  gives  necessarily  y'  zero,  and  y  a  constant.  This  would 
make  the  required  curve  a  right  line,  coinciding  with,  or  par- 
allel to,  the  axis  of  x.  While  this  solution  is  correct  so  far  as 
the  general  form  of  the  required  curve  is  concerned,  it  will 
not  be  always  possible  to  draw  such  a  line  through  two  fixed 
points  given  at  pleasure,  unless  we  are  at  liberty  to  assume 
the  axis  of  x  so  as  to  make  y^  and  y^  equal,  which  is  not  con- 
templated.    We  must,  therefore,  seek  another  solution. 

9.  We  will  begin  by  transforming  ^ y'  thus: 


/     dy 
^      dx 

Change  y  into  y  +  Sy^  while  ;r,  and  consequently  dx,  undergo 


TO  CALCULUS  OF   VARIATIONS. 

no  alteration.     Then  denoting  the  new  value  of  y'  by  Y' ,  we 
have 

Whence,  subtracting  from  the  first  member  y\  and  from  the 

last  its  equal  -^,  we  have 
dx 


V 

-y-- 

dSy 
~  dx' 

t  y 

-y= 

■h'. 

whence 

6/= 

ddy 
dx' 

like 

manner, 

/= 

dy 

dx'' 

Change  y  into  y-^-^y-     Then 


dx'^-^^   -^^       dx'^   dx'' 


Y"—y" 


•^        dx'  ' 
and,  similarly, 

-^  dx"" 

where  n  is  any  positive  integer. 

10.  Equation  (6)  may  now  be  written 

^x^    V  dx 
But  integrating  by  parts,  we  have 


SHORTEST  PLANE   CURVE  BETWEEN   TWO  POINTS.       II 

where  the  suffix  i  denotes  what  the  quantities  affected  become 
when  X  is  x^,  and  o  what  the  same  quantities  become  when  x 
is  x^.  But  since  the  two  points  through  which  the  required 
une  must  pass  are  fixed,  Sy^  and  dy^  are  each  zero ;  that  is, 
y  receives  no  increment  at  these  points,  and  therefore  (8) 
becomes 

This  equation  can  be  satisfied  by  writing 
d  y'  y 

Squaring,  clearing  fractions,  and  transposing,  we  have 

y-  _  cy^  =  c\        /=:  -M=.  =  a,        y  =  ax  +  d, 

Vi  —  ^ 

the  general  equation  of  the  straight  line. 

11.  It  will  be  seen  that  the  solution  y=o  is  only  a  par- 
ticular case  of  the  more  general  one  just  obtained,  and  we  are 
therefore  led  to  inquire  why  the  method  pursued  in  Art.  8 
did  not  give  a  satisfactory  result.     Now,  since  we  have  the 

equations  Sy'  =  — -^,  ^-  =  Cy  (6)  may  be  written 
ax     V 

£?V  tf/  dx  =  £\dSy  =  o, 
whence,  by  integration. 


12  CALCULUS  OF  VARIATIONS. 

and  because  both  dy^  and  8y^  are  zero,  this  equation  can  be 
satisfied  without  making  c  zero. 

The  error,  therefore,  in  Art.  8  appears  to  have  arisen  from 
the  fact  that  we  required  the  curve  to  pass  through  two  fixed 
points,  anii  then  entirely  disregarded  that  condition  in  obtain- 
ing our  solution.  But  (9)  was  estabhshed  by  expressly  impos- 
ing this  condition  upon  the  problem;  and  as  there  are  no 
further  conditions  to  be  imposed,  and  as  ^y  cannot  be  further 
transformed,  that  equation  can  only  be  satisfied  by  equating 
to  zero  the  coefficient  of  dy  dx  in  that  equation. 

12.  Resuming  equation  (5),  let  us  next  consider  the  term 
of  the  second  order, 


Jxa 


''~Sy-dx.  (10) 


If  the  solution  given  above  be  a  true  minimum,  this  term 
must  become  positiv^e,  or  must  reduce  to  zero.  Now  since 
X  is  the  independent  variable,  dx  is  always  supposed  to  be 
estimated  positively  ;  and  as  d/^  can  never  be  negative,  if  we 
also  regard  Fas  positive,  we  see  that  every  element  of  (10)  is 
positive,  and  that  consequently  the  integral  itself  must  be  of 
the  same  sign.  We  conclude,  therefore,  that  a  right  line  is 
the  plain  curve  of  minimum  length  between  two  fixed  points. 

If  the  coefficient  of  Sy'^dx  in  (10),  which  we  may  call  Z, 
could  have  changed  its  sign  within  the  given  limits  of  inte- 
gration— that  is,  if  Z  could  have  been  positive  throughout 
some  portions  of  the  curve,  and  negative  throughout  others 
— we  could  make  (10)  take  either  sign,  and  there  could  be 
neither  a  maximum  nor  a  minimum.  For  by  var3^ing  y' 
throughout  those  portions  of  the  curve  for  which  Z  was 
negative,  while  leaving  the  other  portions  unvaried,  the  inte- 
gral would  become    negative,  or  by  pursuing    an    opposite 


SHORTEST  PLANE    CURVE   BETWEEN   TWO  POINTS.        1 3 

course  it  would  become  positive.  Hence,  in  this  and  similar 
cases,  the  coefficient  of  dy''^dx  must  be  of  invariable  sign  for 
all  values  of  x  from  x^  to  ;r,. 

If  Z  could  have  reduced  to  zero  throughout  the  Avhole 
range  of  integration,  thus  rendering  the  integral  itself  zero, 
we  might  generally  infer  that  the  solution  was  neither  a  maxi- 
mum nor  a  minimum.  For  in  order  to  the  existence  of  either, 
the  term  of  the  third  order  involving  Sy'^  must  also  vanish, 
which  would  seldom  if  ever  occur. 

It  will  be  observed  that  the  term  of  the  second  order  is 
positive  whether  the  extremities  of  the  required  curve  are 
supposed  to  be  fixed  or  not.  But  if  we  disregard  this  con- 
dition, the  terms  of  the  first  order  would  not  vanish,  so  that 
we  would  not  obtain  a  minimum,  except,  indeed,  we  adopt 
the  particular  solution  of  Art.  8.  We  shall,  however,  subse- 
quently show  that  when  the  limiting  values  of  x  only  are 
given — that  is,  when  the  required  curve  is  merely  to  have  its 
extremities  upon  two  fixed  lines  perpendicular  to  the  axis  of 
X — the  solution  of  Art.  8  is  that  which  must  be  taken. 

13.  In  the  preceding  discussion  we  have  merely  proved 
that  the  straight  line  between  two  fixed  points  is  shorter  than 
any  other  plane  curve  which  could  be  derived  from  it  by 
making  indefinitely  small  changes  in  the  inclination  of  its  tan- 
gents to  the  axis  of  x,  either  in  certain  portions  or  through- 
out its  whole  extent.  We  could  not,  therefore,  by  the  use  of 
the  calculus  of  variations  alone,  become  certain  that  the 
straight  line  is  the  shortest  plane  curve  which  can  be  drawn 
between  two  fixed* points,  but  merely  that  it  is  a  curve  of 
minimum  length,  the  existence  of  other  minima  being  possible ; 
one  of  which  might,  perhaps,  be  less  than  the  present,  and 
might  itself  be  the  shortest  curve. 

Again,  the  precpding  method  does  not  permit  us  to  com- 
pare the  straight  line  with  all  other  plane  curves  which  can  be 
drawm  indefinitely  close  to  it.      For  in  developing  /',  Art.  3, 


14  CALCULUS  OF  VARIATIONS. 

we  were  obliged  to  ascribe  indefinitely  small  increments  or 
variations  to  y'  only,  since  y  did  not  directly  or  explicitly 
occur  in  /.  Hence  the  curve  which  we  derive  by  variations 
can  have  no  abrupt  change  of  direction ;  because  no  such- 
change  could  occur  without  rendering  dy'  appreciably  large 
at  that  point.  Therefore  all  curves  with  cusps,  and  all  systems 
of  broken  lines,  are  excluded  from  the  comparison,  although 
it  is  evident  from  the  figure  that  such  curves  might  be  drawn 
without  making  the  variations  of  y  appreciable,  but  only  those 
of/. 


(4.  From  the  remarks  of  the  preceding  article,  which 
were  deemed  'necessary  in  order  to  guard  the  reader  against 
certain  misconceptions  which  are  common  among  students  of 
this  subject,  it  must  not  be  inferred  that  the  calculus  of  vari- 
ations is  of  little  use  as  a  method  of  solving  questions  of  max- 
ima and  minima.  For  we  shall  see  as  we  advance  that  it  can 
in  general  be  made  to  give  a  satisfactory  solution  when  such 
a  solution  exists.  Indeed,  the  recent  discoveries  relative  to 
the  theory  of  discontinuity,  which  are  due  chiefly  to  the  labors 
of  Prof.  Todhunter,  and  of  which  we  shall  speak  hereafter, 
show  that  this  branch  of  the  calculus  does  not  in  reality  fail 
to  present  solutions  even  in  very  many  of  1;fiose  cases  in  which 
its  failure  has  been  hitherto  assumed. 

!6.  It  remains  only  to  determine  the  constants  a  and  b 
which  occur  in  the  general  solution.  It  will  appear  that  since 
the  required  line  is  to  pass  through  two  fixed  points  whose 
co-ordinates  are  x^,  y^,  x^,  jj/„  we  must  have 


BRACHISTOCHRONE  BETWEEN   TWO  POINTS.  1 5 

and  therefore  so  soon  as  these  quantities  are  given  a  becomes 
known.     Then  to  determine  b,  we  have  jKo=  <^^A-  ^^ 

b  =  y  —  ax,=y  —  ^^^^'^  ,r„, 

and  thus  b  is  also  known  when  x^,  x^,  /„,  j\,  are  fully  given. 

16.  In  further  illustration  of  our  subject  we  next  proceed 
to  consider  another  problem,  the  solution  of  which  is  not  so 
generally  known. 


Problem  II. 

It  is  required  to  determine  the  equation  of  the  plane  curve, 
doiun  which  a  particle,  acted  upon  by  gravity  alone ^  would  descend 
from  one  fixed  point  to  another  in  the  shortest  possible  time. 

Let  a  be  the  upper  and  b  the  lower  point.  Assume  the 
axis  of  x  vertically  downward,  and  a  as  the  origin  of  co-ordi- 
nates. Also  let  the  variable  s  be  the  length  of  the  required 
curve  at  any  point  measured  from  a ;  v,  the  velocity  of  the 
particle  at  the  same  point ;  and  /,  its  time  of  descent  from  a  to 
that  point.  Then  we  wish  to  determine  the  curve  which  will 
render  T  a  minimum,  where  T  is  the  total  time  of  descent 
from  a  to  b,  or  what  t  becomes  at  the  point  b.  We  must  first 
then  find  /  as  a  function  of  x  and  j/,  or  their  differentials. 

Now,  from  the  well-known  differential  equations  of  motion 
in  mechanics,  we  have 

dt^'^'  (I) 

v 

and,  Art.  2, 

ds  =  Vdx'  +  d/=  Vi  +/'  dx. 


1 6  CALCULUS   OF  VARLATIONS. 

We  also  know  that  the  particle  loses  no  velocity  in  pass- 
ing from  one  point  to  another  of  a  curve  with  no  abrupt 
change  of  direction,  and  that  therefore,  if  it  start  from  a 
state  of  rest  at  a,  its  velocity  at  any  point  of  the  curve  must 
equal  that  which  it  would  have  acquired  in  falling  freely 
through  the  same  vertical  distance.     Hence  we  shall  have 

V  =  V2gXy 

g  being   the   acceleration    due    to    gravity.      Therefore   (i) 
becomes 


y2gx 


and 


T=  /         , ~       dx,  (2) 


which  is  to  become  a  minimum. 

17.  But  since  ^  is  a  constant,  the  second  member  of  (2) 
may  be  written 


^'  ^'  +^"  d.. 


V2g  ^^^  Vx 

Now,  it  is  evident  in  general  that  if  c  times  any  integral  is  to 
be  a  maximum  or  a  minimum  (c  being  any  constant),  the  in- 
tegral itself  must  also  be  a  maximum  or  minimum.  Hence, 
omitting  the  constant  factor,  the  expression  to  be  rendered  a 
minimum  in  this  problem  may  be  written 


U=  f   — -"t^"  dx  =  n  Vdx,  (3) 

Now,  as  in*  the  preceding  problem,  change  y  'mX^o  y'  -\-^y' 
and  develop  by  Taylor's  Theorem.     Then  we  shall  obtain 


BRACHISTOCH-RONE    BETWEEN   TWO  POINTS.  1 7 

=    /     ——J=Sy'dx^    /      :=. •   dyV4r  + etc.      (4) 

We  shall  not  in  future  develop  any  variation  beyond  the 
terms  of  the  second  order,  since  if  the  terms  of  the  first  two 
orders  should  become  zero,  there  could  rarely  if  ever  be 
either  a  maximum  or  a  minimum,  as  explained  in  Art.  12. 

Hence  we  must  have 

r^  —=jL=Sy'dx  =  o,  (5) 

But  since  the  two  extreme  points  are  fixed,  we  must  impose 
this  condition  upon  the  problem  by  integrating  (5)  by  parts, 
as  in  the  preceding  problem,  and  neglecting  the  terms  thus 
freed  from  the  integral  sign,  because  containing  6y^  and  dy^. 
Performing  this  operation,  we  shall  obtain 


-  r'~--=jL=    dydx^o,  (6) 

=  0,  (7) 

=  c.  (8) 


dx  \/^{^i  _|.y2) 

d         y 

y 


Vx{i  +y^) 

Now  since  c  in  the  last  equation  is  an  arbitrary  constant,  make 


1 8  CALCULUS  OF  VARIATIONS. 

it  equal  to  -=.     Then  squaring,  clearing  fractions,  and  trans- 

^  a        ■      . 

posing,  we  have 

'-—  'J^  —  £ 

a         a 
Whence  solving  fory,  we  obtain 

y  =     . >  (9) 

Va  —  X 

which  is  known  to  be  the  differential  equation  of  the  cycloid. 
Therefore 

J/  =  versin -^^ x  ~  Vax  —  x^-\-  b,  (lo)  x 

where  a  is  twice  the  radius  of  the  generating  circle,  and  b  is 
zero,  because  the  origin  was  taken  at  the  upper  point.  The 
last  equation  may  be  finally  written  thus : 

j  =  r  versin"^ \^2rx  —  x'',  (ii) 

where  the  circular  function  is  natural,  and  r  is  the  radius  of 
the  generating  circle. 

18.  By  disregarding  the  condition  that  the  curve  must 
pass  through  the  two  fixed  points,  we  shall,  as  in  the  preced- 
ing problem,  obtain  from  (5), 

y' 

z=:0, 


VxiT^y'^) 

which  makes  y'  zero,  and  y  a  constant,  which  must  also  be 
zero,  because  the  curve  passes  through  the  origin.  There- 
fore the  curve  would  in  this  case  coincide  throughout  with^ 


BRACHISTOCHRONE  BETWEEN   TWO  POINTS.  1 9 

the  axis  of  x,  which  solution  could  only  be  possible  when  the 
two  points  were  in  the  same  vertical  line,  and  then  its  truth  is 
self-evideht. 

19.  Let  us  now  consider  the  term  of  the  second  order,  viz.. 


Jxa 


^sy^dx,  (12) 


^0    2|/;r(l+y7 

If  the  cycloid  be  the  true  solution  of  our  problem,  this  term 
must  become  positive,  whether  y'  be  varied  throughout  the 
whole  integral  or  only  throughout  certain  portions  taken  at 
pleasure.  To  satisfy  this  condition  it  is  merely  necessary  that 
Z,  the  coefficient  of  Sy'dx  in  (12),  shall  become  positive  and 
not  change  its  sign  as  we  pass  from  a  to  b.  But  since  x  can- 
not become  negative  in  this  problem,  the  square  root  of  x  is 
real  and  may  be  considered  as  always  positive  from  a  to  b  \ 
then,  as  we  may  regard  Vi  -\- y'"^  as  always  positive,  the  above 
conditions  are  satisfied,  and  we  conclude  that  the  cycloid, 
having  a  cusp  at  a,  its  base  horizontal,  and  its  vertex  down- 
ward, is  a  solution  of  our  problem. 

Let  us  also  try  the  solution  /  =  o  of   Art.  18  ;  this  will 
reduce  (12)  to 


J. 


•"•    2Vx 

which  will  also  become  necessarily  positive  if  we  assume  \G: 
to  be  positive.  Thus  this  solution  likewise,  when  it  is  pos- 
sible, renders  T  a  minimum,  as  it  evidently  should. 

20.  Remarks  similar  to  those  made  in  Art.  13  apply  also 
to  this  example.  For  it  is  plain  that  we  have  only  compared 
the  cycloid  as  a  curve  of  descent,  with  all  other  curves  pass- 
ing through  the  given  points,  having  no  abrupt  change  of 
direction,  and  drawn  indefinitely  near  to  it.     Hence  we  have 


20  CALCULUS  OF  VARIATIONS. 

in  reality  only  shown  that  the  cycloid  is  one  of  the  curves 
which  renders  T  a  minimum,  the  term  minimum  being  used 
in  the  technical  sense  hitherto  explained.  However,  as  in  the 
fornier  problem,  these  restrictions  are  merely  theoretical,  and 
are  noticed  in  order  to  prevent  misconceptions  which  might 
occasion  difficulty  in  subsequent  discussions. 

For  in  the  present  case  the  cycloid  between  two  points  is 
undoubtedly  the  curve  of  quickest  descent  from  one  to  the 
other,  and  from  this  property  it  is  often  called  the  brachisto- 
chrone. 

21.  In  addition  to  what  has  been  already  said,  we  must 
here  call  attention  to  another  point  which  is  often  passed 
over  by  elementary  writers  on  this  subject.  Suppose  y'  to 
become  infinite  for  some  point  within  the  range  of  integra- 
tion, as  it  does  at  the  vertex  of  the  cycloid.  Then  when  we 
change  y  intoy  -\-  Sy' ,  if  we  regard,  as  we  must  by  the  theory 
of  the  subject,  Sy'  as  taken  arbitrarily,  but  always  indefinitely 
small,  we  can  make  the  new  or  derived  curve  assume  any 
form  we  please,  except  that  its  tangent  at  X  must  have  the 
same  direction  as  that  of  the  cycloid  at  the  vertex,  where  X 
IS  the  abscissa  of  the  vertex.  For  suppose  the  vertex  tangent 
of  the  cycloid  to  undergo  a  slight  change  of  direction,  so  that 
its  new  angle  of  inclination  to  ;i'  may  differ  from  a  right  angle 
in  an  indefinitely  small  degree.  Then  we  cannot  assert  that 
this  small  change  of  direction  could  be  produced  by  an  in- 
definitely small  change  in  the  value  of  y\  or  the  natural  tan- 
gent of  the  right  angle.  That  is,  owing  to  the  indeterminate 
nature  of  infinity,  we  cannot  with  certainty  apply  the  method 
of  variations  to  any  element  of  the  integral  which  is  affected 
by  an  infinite  value  of  y' ,  and  hence  the  integral  must  not  be 
extended  so  as  to  include  this  element.  In  the  present  case, 
then,  we  are  only  sure  of  a  minimum  so  long  as  we  are  not 
obliged  to  go  beyond  the  vertex  of  the  cycloid  for  b. 

But  the  occurrence  of  an  infinite  value  of  y'  in  any  case 


PLANE  BRACHISTOCHRONE-  BETWEEN  TWO  POINTS.       21 

will  not  warrant  us  in  concluding  that  the  solution  does  not 
give  a  true  maximum  or  minimum,  even  when  the  integral 
includes  that  value  of  y'.  All  that  we  can  say  is  that  the  pro- 
posed method  becomes  inapplicable.  Indeed,  we  shall  have 
occasion  to  show  that  sometimes,  by  changing  to  polar  co- 
ordinates, or  by  some  other  change  of  the  independent  vari- 
able, the  integral  may  in  these  cases  be  freed  from  infinite 
quantities,  and  the  previous  solution  shown  to  give  a  true 
maximum  or  minimum. 

Of  course  if  we  regard  Sy'  as  zero  when  y'  becomes  in- 
finite— that  is,  consider  the  tangent  to  the  curve  as  fixed  at 
that  point — the  variation  of  the  element  becoming  zero,  may 
be  included  in  the  development,  and  all  difficulty  disappears. 

It  will  be  observed  that  F becomes  infinite  at  A,  and  the 
solution  is  therefore  still  subject  to  any  objection,  but  there 
would  seem  to  be  none,  which  can  arise  from  this  fact. 

Case  2. 

22.  As  a  means  of  still  further  extending  our  knowledge 
of  variations,  let  us  resume  the  preceding  problem,  merely 
taking  the  horizontal  as  the  axis  of  x. 

Then,  the  notation  and  the  other  conditions  being  un- 
changed, we  must,  as  before,  render  T  a  minimum.  But,  as 
formerly, 


ds 


dt  =  -^,  ds  =  Vdx'+d/  =  Vi  +y'dx, 


V 


where  y'  now  means  the  natural  tangent  of  the  angle  which 
any  tangent  to  the  curve  makes  with  the  horizontal  instead  of 
the  vertical  axis.  Also,  v  =  V2gy,  so  that,  neglecting,  as  be- 
before,  the  constant  factor,  we  must  minimize  the  expression 


22  CALCULUS  OF  VARIATIONS. 

Now  in  V  change  y  into  y  +  dj,  and  /  into  /  +  Sy' .  Then 
we  may  develop  V ,  or  the  new  state  of  F,  by  the  extension 
of  Taylor's  Theorem,  thus : 


We  also  have 


u'  ^  r'v'dx, 

where  U'  is  what  ^becomes  when  we  change  Finto  V-\-dV 
or  into  V,  dx  being  unaltered.  Hence  caUing  U'  —  U,  dU, 
we  have 

Indeed,  it  is  evident  that  a  similar  course  could  be  pursued 
should  V  contain  any  number  of  quantities  capable  of  being 
varied. 

23.  It  may  be  well  before  proceeding  further  to  refer 
briefly  to  the  subject  of  maxima  and  minima  of  functions 
involving  more  than  one  variable,  as  it  is  developed  by  the 
differential  calculus. 

Let  /  be  a  function  of  x,  y,  z,  etc.  Give  small  increments, 
k,  i,  k,  etc.,  to  X,  y,  z,  respectively,  and  develop  /',  the  new 
state  of  /,  by  Taylor's  Theorem.  Then  the  terms  of  the  first 
order  in  /'—/will  be 

^^  I  ^/;4_^/ ^j_etc 


PLANE    BRACHISTOCHRONE    BETWEEN   TWO  POINTS.     23 

which  must  collectively  vanish ;  and  if  the  quantities  h,  i,  k, 
etc.,  be  independent,  each  of  the  partial  differential  coeffi- 
cients of  /  must  also  vanish.  Then  the  terms  of  the  second 
order. 


2  \dx^  dxdy        '    dy^ 

must  become  collectively  negative  for  a  maximum  and  posi- 
tive for  a  minimum.  Also,  if  the  increments  be  independent, 
the  second  partial  differential  coefficients  of  /  must  fulfil  cer- 
tain conditions  among  themselves,  for  an  account  of  which,  as 
they  have  no  application  here,  the  reader  is  referred  to  works 
on  the  differential  calculus. 

24-.  The  expression  for  SUm.  (i)  is  similar  to  that  for/'—/, 
only  each  term  is  multiplied  by  dx,  and  is  under  an  integral 
sign,  Sy  and  Sy'  taking  the  place  of  h  and  i,  dx  being  regarded 
as  constant.  In  the  present  case,  therefore,  the  two  integrals 
of  the  first  order  in  (i)  must  collectively  vanish,  while  the  three 
integrals  of  the  second  order  must  become  collectively  positive. 

25.  We  have 

dV _        Vi+y  dV _  y'  d'V  _3Vi+/' 

dy  ~  2y^        '          dy'  ~   ^{^  -\- y''')y  df   ~~  4/^ 

d^v  ^  y      -       ^r^        I 

dydy'  2  \/\i  +/V         dy""        |/(i  4-/7/ 

Hence  equation  (i)  becomes 


A — Sy  \  dx.  (2) 


24  CALCULUS  OF  VARIATIONS. 

Whence  we  have 


Now,  it  might  at  first  appear  that  we  could  regard  8y  and 
dy'  as  independent,  and  thus  might  equate  to  zero  each  of  the 
integrals  in  (3).  But  since  the  curve  is  to  pass  through  two 
fixed  points,  this  condition,  which  has  not  yet  been  regarded, 
must  be  imposed  upon  the  problem,  and  may  be  said  to  limit, 
in  some  sense,  the  independence  of  ^y  and  Sy\  This  condi- 
tion can  be  imposed  by  means  of  the  second  integral  only, 
since  the  first  is  incapable  of  any  further  integration.  For 
putting  for  dy'  its  value  from  (A),  we  have 

4/(1 +y>^-^  ^(i+y>  ^  dx  ^{^ij^yy  •" 

Hence,  since  Sy^  and  Sy^  are  zero,  when  we  make  the  integral 
definite,  the  two  terms  which  will  be  without  the  sign  of  in- 
tegration will  disappear,  and  we  shall  have 

and  therefore  (3)  may  be  written 


_rJi^L3^  +  -^_-/_U^^.^o.  (4) 

t/o^o    I        2y^  dx   ^^{j^j^yyS 

Thus  6y'  has  been  eliminated,  and  there  being  no  further 
conditions  to  impose;  (4)  can  only  be  satisfied  by  writing 


^•+^%^ -..4=^  =  0.  (5) 


27^  dx   ^i^iJ^yy 

Multiply  the  first  term  by  dy,  and  the  second  by  its  equal, 
ydxj  and  we  have 


PLANE    BRACHISTOCHRONE    BETWEEN   TWO  POINTS.      2$ 


^=^dy-\-y'- — •"-  dx  =  o,  (6) 

Then  by  parts, 

and  again  by  parts, 

so  that  we  have,  finally, 


—      -^ T^~  —  a  constant,  say  -—^,         if) 

|/(i  +y^)j/  y  ■       ^   Va   .      ^^ 

Now  reducing  the  first  member  to  a  common  denominator, 
Ave  have 

=  ^,     y{i+y^)  =  a,     y^  =  ^^;     (8) 


1/(1  +y^)j^       |/^  J 

which  last  equation  cannot  be  integrated  by  solving  for  y. 
But  we  readily  obtain 

I  dx  Vy 

-7    or     -—  =  -— -•^— , 
ji/  dy         s/a—y 

which  is  as  before  the  differential  equation  of  the  cycloid,  in 
which  a  equals  2r ;  only  x  and  y  have  been  interchanged,  as 
will  appear  from  equation  (lo).  Art.  17. 

26.  If  we  disregard  the  condition  that  the  curve  is  to  pass 
through  two  fixed  points,  we  shall  have,  from  (2), 


26  CALCULUS   OF  VARIATLONS. 


Now  the  first  of  these  equations  can  only  be  satisfied  by 
equating  to  zero  the  coefficient  of  dydx,  and  then,  as  we  may 
evidently  neglect  the  supposition  that  y  is  infinite  throughout 
the  curve,  we  have,  necessarily, 


s/i  +/'  =  0,     y=±  V^^  ; 

a  result  which  shows  that  a  solution  by  this  method  is  impos- 
sible. 

The  solution  y  =  o  of  Art.  i8,  which  will  become  in  this 
case  J/'  =  CO ,  is  also  suggested  by  this  method ;  for  if  in  the 
second  of  equations  (8)  we  make  a  infinite,  then,  since  y  cannot 
be  always  infinite,  we  shall  find  that  y  is  infinite.  This  solu- 
tion, representing  the  vertical  through  A^  has  been  already 
shown  to  give  a  true  minimum  ;  although  the  considerations 
of  Art.  20  show  that  it  could  not  be  investigated  so  long  as 
the  horizontal  is  taken  as  the  independent  variable.  This  case 
then  exemplifies  the  remarks  there  made  relative  to  over- 
coming, by  a  change  of  the  independent  variable,  the  difficulty 
presented  by  the  occurrence  of  infinite  quantities. 

27.  Let  us  now  examine  the  sign  of  the  terms  of  the 
second  order  in  dU.  Since  those  of  the  first  order  vanish,  we 
have,  from  (2), 


^-'£■1 


sy      -^2x^1  +yy 


H /.      ,      ,^^^   c  dx,    (10) 


From  the  second  of  equations  (8)  we  have 

|/yT  +  yr)=  ^a, 


PLANE    BRACHISTOCHRONE    BETWEEN   TWO  POINTS.     2y 

and  therefore  (lo)  becomes 

6U  =S;  i  'l§Sf  -  -4.  6y  Sy  +  -^  Sy^  I  d..  (I  I) 

^^°     (    oy  2y  Va  2a  Va         j 

But  ?—  can  be  written ^ ,  where  r  =  -,  or  the  radius 

8/'  2  Va .  2/  2 

of  the  generating  circle.     Whence  (i  i)  becomes 

6U=-^r\^,S/-^-dy  6y'  +  I  dy'^  \  dx.         (12) 

But,  from  equation  (A), 

//  Sy6yd.=l'-l-r-l4.y-.cl..  (13) 

«/j-^-^  y    2       ^    2    dx  y  ^ 

Put/ for  •^'.      Then 

y 

£y  ,y  Sya,  ^  1  [i^  s/,  -  /,  Sf]  -  j/r^/^^^.      (14) 

But  since  the  extreme  points  of  the  curve  are  fixed,  dy^  and 
6y^  are  each  zero,  and  we  have 

£y  sy  sydx  =  -  '-Xyy  ^  dx.  (IS) 

But 

^dx^di^  y^y'  -/'^y  ='^-t-yLdx; 

dx  y  y         y^ 


and  because  dy  =  y'dx,  the  last  equation  may  be  written 

^^  ^^  -  j  I  y'dy    y 

dx  \y    dy         f 


^^dx=\iy^-Qdx.         (16) 

[y    dy  f  ) 


28  CALCULUS  OF  VARIATIONS. 

Now  differentiating  the  third  of  equations  (8)  and  dividing  by 
2,  we  have  , 


.r          ^ 

2/ 

f 

and  therefore 

(i6)  becomes 

dl    , 

—  dx=- 

dx 

-\-+'~ 

\/^/ 

•  dx. 

But,  from  the  third  of  equations  (8), 

whence 

y"-- 

2r  —  y 

~    y    ' 

Therefore 

dl  , 

--dx:= 

dx 

dx. 

(17) 


.  ly^6y8ydx  =  \lyf^JL^dx.  _         (18) 

Substituting  this  value  in  (12),  we  have,  finally, 

and  it  is  evident  that  this  integral  is  positive,  since  each  of  its 
elements  is  positive. 

28.  Although  we  might  infer  from  the  preceding  article 
that  we  have  a  minimum,  that  term  being  used  in  its  technical 
sense,  still  our  investigation  will  not  be  entirely  trustworthy 
unless  we  regard  the  direction  of  the  tangent  at  A  as  absolutely 

fixed.     For  we  have  seen  that  y  =    ^  —  y ^  ^^^  therefore  when 

y 


PLANE    BRACHISTOCHRONE    BETWEEN   TWO  POINTS.      2g 

y  is  zero,  y'  becomes  infinite.  That  is,  we  cannot  with  confi- 
dence include  in  our  investigation  every  element  of  the  definite 
integral  U,  because  at  ^,  F  becomes  infinite.  We  cannot, 
however,  conclude  that  there  is  not  a  minimum,  because  we 
do  not  know  what  effect  a  variation  of  y  in  this  element  would 
have  upon  the  general  result.  Indeed,  we  do  know  that  if  the 
second  point  be  not  beyond  the  vertex,  we  have  a  true  mini- 
mum, and  we  now  see  also  that  if  the  tangent  at  A  be  fixed — 
that  is,  if  the  cycloid  be  compared  with  any  other  derived 
curve  whose  tangent  is  at  right  angles  to  the  horizontal — we 
shall  in  any  case  have  a  minimum. 

The  term  derived  will  be  used  to  denote  any  curve  which 
can  be  obtained  from  the  original  or  primitive  curve  by  the 
method  of  variations,  and  must  therefore  be  always  indefinitely 
near  to  its  primitive,  and  without  abrupt  change  of  direction. 

29.  The  preceding  discussion  shows  the  advantage  of 
taking  the  vertical  as  the  independent  variable.  For  while 
the  result  by  either  method  is  the  same,  as  indeed  it  must  be 
in  every  case,  it  is  much  more  easily  obtained  by  the  former. 
This  is  due  to  the  fact  that  in  the  former  case  x,  being  inca- 
pable of  variation,  enters  the  function  F,  thus  leaving  y  only 
to  be  varied,  while  in  the  latter  y  and  /,  both  being  capable 
of  variation,  enter  V,  thus  rendering  the  problem  one  of  two 
variables. 

When  we  come  to  the  terms  of  the  second  order,  the 
results  apparently  agree  also.  But  while  that  in  the  former 
case  is  readily  obtained,  and  is  probably  entirely  trustworthy 
so  long  as  we  do  not  wish  to  pass  the  vertex,  in  the  latter 
case  some  transformation  is  required  in  order  to  obtain  any 
result,  and  even  then,  owing  to  the  occurrence  of  an  infinite 
value  of  y  at  the  outset,  we  cannot  rely  implicitly  upon  our 
investigation  unless  we  regard  the  derived  curve  as  having  at 
A  the  same  tangent  as  its  primitive ;  that  is,  the  vertical. 


30  CALCULUS  OF  VARLATIONS. 

Problem  III. 

30.  //  is  required  to  determine  the  form  of  the  plane  curve 
which  shall  pass  through  two  fixed  points^  and  which  shall  include 
between  itself ^  its  evolute,  and  its  radii  of  curvature  at  the  two 
fixed  points  a  minimum  area  ;  the  extreme  tangents  of  the  required 
curve  being  also  fixed. 

As  before,  let  ds  be  an  element  of  the  required  curve,  r  the 
radius  of  curvature,  and  U  the  area  which  is  to  become  a 
minimum.     Then 

U=f\ds,  (I) 

and  we  must  first  express  U  in  terms  of  x,  y,  y,  etc. 
We  have 

ds  =  Vi  +/'  dx, 


^_(dx^  +  dyy^  d/  _  ds^  _V{i+yy      ,. 

dxdy  dxdy       y"dx'  y"         '  ^^ 

the  sign  ±  having  been  disregarded.  Substituting  these 
values,  and  assuming  that  the  curve  is  to  be  concave  to  the 
axis  of  X,  and  y"  therefore  negative,  (i)  may  be  written 

Now  change  y'  into  y'  -f-  Sy\  y"  into  y"  -\-  Sy\  and  develop  as 
before.  Then  including  the  terms  of  the  second  order,  we 
have 


2t/xo      ( 


y  y 


V^^^^f^]^^.    (4) 


THE   E  VOLUTE   PROBLEM.  31 

Whence  we  must  have 

Now  it  is  plain,  as  before,  that  the  two  integrals  combined 
in  the  last  equation  are  not  independent,  there  being  here  two 
conditions  to  be  imposed  upon  the  problem  ;  namely,  that  ^}\ 
and  ^y^  shall  vanish,  and  also  that  Syl  and  Sy^  shall  vanish. 

To  impose  these  conditions,  we  have  only  to  extend  the 

method  already  employed.     Thus,  putting  K  for    -^  ^    „         -, 
we  have 

/  K  Sy'dx  :=  K  dy  —  I   "--—-  6y  dx, 
/;■  K  Sy-d.  =  /r.  6y,  -  a;  Sy„  -  ly^Sy  dx.  (6) 

Also  putting  L  lor  ^       „•;    ' ,  and  observing  .that 

y 

^    ~   dx'    ~  ~dx  ' 
we  have 

And  in  a  similar  manner  we  obtain 

r  'jL  Sy'd.  =  m  6y^  -  m]  6y^  -    T'  ^  6y  d.. 


32  CALCULUS   OF    VARIATIONS. 

Collecting  and  arranging  these  results,  (5)  becomes 

Now,  if  we  suppose  Sy^,  dj//,  ^j/^,  Sj//,  to  severally  vanish, 
we  shall  thereby  impose  the  two  given  conditions  upon  the 
problem,  and  (8)  will  become 

As  there  are  no  further  conditions  to  impose,  this  equation 
can  only  be  satisfied  by  writing 

Restoring  the  values  of  K  and  Z,  and  integrating,  we  have 

y"  ^  dx       f'        -Vc-o,  (II) 

which,  since  dy'  z=z  y"dx,  may  be  written 

^l(LpylUy^y,,Sl±^  +  cdy'  =  o.  (12) 

Then  integrating  by  parts,  we  have 


THE  E  VOL  UTE  PROBLEM.  3 3 

Hence  (12)  gives 

iSL^yy^cy+c'=.o,  (13) 

(I  +yy  ^  4; 
But  from  equations  (i),  (2)  and  (3)  we  readily  obtain 


r 


_      dx 

and  substituting  this  value  in  (14),  observing  t\i2itydx  =  dy, 
we  have 

Let  t  denote  the  angle  which  the  tangent  to  the  required 
curve  at  any  point  makes  with  the  axis  of  x.     Then  ~  =  sin  /, 

dx 
and  -J-  =  cos  t.     Also,   let  d   be  the  constant   angle  whose 

ds 

natural  tangent  is  -.     Then  c  =  h  sin  b,  and  c'  =  h  cos  b ;  // 

being  some  constant  at  present  unknown.     Then  substituting 

in  {15)  these  values  of  c,  dy  ~,  — -,  it  becomes 

ds    ds 

r  =.  —  (sin  /  sin  ^  -f-  cos  t  cos  ^)  =  -  cos  (/  —  B),        (16) 

which  is  the  intrinsic  equation  of  the  cycloid,  h  being  equal  to 
eight  times  the  radius  of  the  generating  circle,  and  b  the  angle 
'made  with  the  axis  of  x  by  the  chord  joining  the  cusps. 

31.  Let  us  next  examine  the  sign  of  the  terms  of  the  second 
order.     Since  those  of  the  first  order  vanish,  (4)  becomes 


34  CALCULUS  OF    VARIATIONS. 


-  4/  ^-  Sy'Sf  +  ('  +/")'  y-  }  dx 

But  since  the  axis  of  x  is  so  taken  as  to  render  the  cycloid 
concave  to  it,  y"  is  always  negative,  and  therefore  the  factor 

— ~  is  always  positive,  since  the  sign  of  each  element  depends 

y 

upon  that  of  this  factor.  We  infer,  therefore,  that  the  cycloid 
is  the  curve  required  ;  although,  because  y"  becomes  infinite  at 
the  two  cusps,  our  investigation  will  perhaps  be  subject  to 
some  doubt  if  we  are  obliged  to  include  either  cusp  within 
the  range  of  integration. 

32.  If  we  attempt  to  neglect  the  two  conditions  which  are 
to  hold  at  the  limits,  and  to  regard  6y'  and  Sy"  as  independent, 
we  shall  have  the  two  equations 

4/(i  +yo  _  ^      (i+yT  _  ^ 

both  of  which  give  y'^  =  00,  which  cannot  be  true  if  the  re- 
quired curve  is  to  be  continuous.      The   seeming  solution, 


THEORE  TICAL  CONSIDER  A  TIONS.  3  5 

y'  =  o,  of  the  first  equation  must  be  rejected,  because,  if  it 
could  hold,  the  curve  becoming  a  straight  line  would  cause  j^' 
to  vanish  also,  and  thus  the  equation  would  become  indefinite. 

33.  It  is  evident  that  the  cycloid  will  not  give  the  least 
possible  value  of  the  area  in  question.  For  by  joining  arcs  of 
cycloids,  or  even  of  circles,  of  indefinitely  small  radius,  the 
area  may  be  made  as  small  as  we  please,  as  will  appear  by  the 
subjoined  figures : 


We  have,  therefore,  theoretically  only  a  minimum  in  the  tech- 
nical sense  hitherto  explained. 

In  fact,  the  method  here  employed  excludes  all  curves 
having  either  y  or  y'  infinite  within  the  given  range  of  inte- 
gration ;  and  it  also  enables  us  to  compare  the  cycloid  with 
such  curves  only  as  can  be  derived  from  it  by  any  arbitrary 
indefinitely  small  changes  in  the  values  of  y  and  y.  Still, 
under  the  conditions  which  we  imposed  upon  the  problem — 
viz.,  that  the  extreme  points,  and  also  the  direction  of  the 
extreme  tangents,  should  be  fixed,  and  the  subsequent  condi- 
tion that  the  required  curve  should  be  concave  to  the  axis  of 
X — there  can,  we  think,  be  no  doubt  that  the  cycloid  gives  not 
only  a  minimum,  but  also  the  least  value  of  the  area  in  ques- 
tion. 


Section  II. 

CASE  IN  WHICH  THE  LIMITING  VAL  UES  OF  X  ONL  Y  ARE  GIVEN. 

34.  The  reader  having  now  become  somewhat  familiar 
with  the  general  method  of  the  calculus  of  variations,  we  shall 
next  present  some  theoretical  considerations,  which  are  usually 
advanced  before  the  discussion  of  problems  is  attempted. 


^\ 


36  CALCULUS  OF   VARIATIONS. 

Suppose  we  wish  to  determine  the  conditions  which  will  render 
U  a  maximum  or  fninimumy  where 

Then  it  will  be  found,  as  in  the  preceding  examples,  that  U 
can  be  reduced  to  the  form 


U 


:  r^vdx, 


where  V  is  some  function  of  x,  y,  y' ,  y'\  etc. 

Now  change jF  intoj/  +  Sy,  /  into/  +  6y\  etc.,  x  remaining 
unaltered.  Let  V,  in  consequence  of  these  changes,  which  are 
indefinitely  small,  become  V\  and  U  become  W,  Then  we 
shall  have 

U'=  r'V'dx. 

tlXo 

Also  let  U'  -  Uhe  denoted  by  dl/,  and  F'  -  Fby  ^V.  Then 
if  Sy,  sy,  etc.,  be  indefinitely  small,  ^Uand  ^Fwill  also  be  in- 
definitely small.     It  is  clear  also  that  we  shall  have 

U'  -U  or  dU=  r^V'dx-  r^Vdx 

=  r\  v  -v)dx=  r^svdx,         ( I) 

Now  if  we  develop  (^Fby  Taylor's  Theorem,  it  becomes 

-^1(a  df  J^  2  B 6y  Sy  +  CSy'-\-  zDSy  6y"  +  etc.),      (2) 


THEORETICAL    CONSIDERATIONS,  37 

in  which  — ,  — >,  etc.,  are  the  partial  differential  coefficients  of 

dy    ay 

F  with  respect  to  y,  y\  etc. ;  and  A,B,  C,  D,  etc.^  are  the  second 
partial  differential  coefficients  of  V  with  respect  to  the  quan- 
tities whose  variations  immediately  follow  them.  Substituting 
this  value  of  dFin  (i),  it  becomes 

_|_  \fJ\A  S/^2B  dy  dy'  -{.  C  dy" -^  2  D  dy  dy"  +  etc.)  dx.    (3) 

Now,  by  our  previous  reasoning,  the  first  Integra]  must 
vanish  for  either  a  maximum  or  a  minimum,  while  the  second 
integral  must  become  negative  for  a  maximum  and  positive 
for  a  minimum. 

35.  It  has  probably  been  observed  that  our  treatment  of 
the  terms  of  the  first  order  has  been  quite  uniform,  while  our 
treatment  of  those  of  the  second  order  has  differed  in  nearly 
every  case.  The  general  discussion  of  this  latter  part  of  the 
problem,  or,  as  it  is  called,  the  discrimination  of  maxima  arid 
minima,  is  the  most  difficult  of  all  the  subjects  connected  with 
the  calculus  of  variations.  Although  the  foundations  had  been 
laid  by  Legendre  and  Lagrange,  and  the  problem  could  be 
solved  in  certain  cases,  still  no  general  method  was  known 
prior  to  the  year  1837,  when  Jacobi  published  a  theorem, 
which  we  shall  explain  hereafter,  and  which  reduces  this 
portion  of  our  investigation  also  to  a  uniform  rule. 

We  shall,  therefore,  at  present  speak  of  dU  2iS  involving 
terms  of  the  first  order  only,  except  when  the  contrary  is 
expressly  stated. 

36.  Let  us  now  consider  more  generally  than  hitherto  the 
equation  6^^=  o. 


38  CALCULUS  OF   VARIATIONS, 

By  (3)  this  becomes 

and  this  equation  is  true  whether  the  values  of  y,  y,  y" ,  etc., 
at  the  limits  are  fixed  or  not,  it  being  merely  required  that  the 
limiting  values  of  x  only  should  be  fixed.  Now  by  means  of 
the  known  relations  given  in  formulae  (A),  (B)  and  (C)  we 
can,  by  integration  by  parts,  transform  any  term  in  (4)  until  it 
shall  consist  of  terms  free  from  the  sign  of  integration,  and  an 
integral  involving  dy  dx. 

Let  N,  P,  Q,  R,  S,  etc.,  be  the  coefficients  of  Sy,  Sy\  Sy\  etc., 
in  (4),  and  consider  for  example  the  term 

«y.  =^, 

We  have 

^      dx'  dx^        ^  dx  dx^ 

rdSd'dy_       _  _  dSd'dy       Pd'Sd'dy  ^ 
"J  dx  'dx'        '~       dx  dx"  "^^  dx'    dx"        ' 

rd'Sd^      ^  _  cPSdSy  _    rd'Sddy     ^^ 
J  dx""    dx""         ~  dx'   dx       ^  dx'   dx 

^  dx'  dx  dx'    -^    '  ^   dx' 

rs6/^d.=(ssr-f6y"+gsy-g,sy 


xq       -^  \     -"         dx  dx""  dx' 


+  /      -r-r  ^y  dx. 


THEORETICAL   CONSIDERATIONS.  39 

Integrating  the  other  terms  in  (^^  in  a  similar  manner,  collect- 
ing and  arranging  the  results,  we  have 

^^=r-^+^^--^3-+etc.w 


/^      dQ  ,   d'R      d'S  .         \  . 


,  /^      dR  .  d'S        ^   \  .    , 

(^      dR  .  d'S        M  .   / 
-[^-d^  +  d?--''''l'^^ 


+{^-S+^4"-^^^^-(^-S+^^^-)^^»'' 


+  (S-  etc.),  c^j^/"  -  (S  -  etc.)o  <^Jo"'  +  etc. 


in  which  -j-,  -y^,  etc.,  are  the  total  differentials  of  these  quan- 


dP  (PQ 
dx''  dx^ 
titles  with  respect  to  x. 

Finally,  for  convenience,  (5)  may  be  written  thus 


SU=L-£^'Mdydx,  (6) 

and  this  equation  holds,  whether  the  values  of  Sy^  6/,  dy" ,  etc., 
at  the  limits  vanish,  as  we  have  hitherto  supposed,  or  not;  the 
limiting  values  of  x  only  being  required  to  remain  fixed. 


40  .  CALCULUS  OF    VARLATLONS. 

37.  We  see  that  SU\n  (6)  consists  of  two  classes  of  terms 
which  are  essentially  different ;  the  first  depending  solely  upon 
the  values  which  the  quantities  (^J/,  ^y',  etc.,  and  P,  Q,  R,  etc., 
with  their  total  differential  coefficients,  may  have  at  the  limits ; 
while  the  second  is  an  integral  involving  the  general  values  of 
these  quantities.  Now  since  (5^ t/ must  vanish  when  ^is  to  be 
a  maximum  or  a  minimum,  let  us  consider  these  two  parts  of 
(^  ^  separately  in  this  case. 

Write,  for  convenience,. 

L  =  h,  dy-  K  ^jo+  \  ^y:-  k  h:-\-j.  h:-j\  ^y:+  etc.  (7) 

Then  it  is  plain  that  the  several  quantities  Sy^^  6y^,  Sy^,  Sy^\ 
etc.,  are  entirely  in  our  power ;  that  is,  we  may  impose  at  the 
limits  any  conditions  we  please,  so  long  as  all  the  variations 
are  indefinitely  small  and  x^  and  x^  remain  immutable.  It  is 
likewise  clear  that  the  quantities  h^,  h^,  i^,  i^,  etc.,  are  not  in  our 
power.     For  suppose  the  equation 

dU=  L  -{-fj  Mdydx  =  0  (8) 

to  have  been  solved  so  as  to  give  j/  as  a  function  of  x,  say/(;ir). 
Then  this  equation  would  be  a  solution  of  the  problem  to  find 
the  value  of  y,  or  the  equation  of  a  plane  curve,  which  would 
render  Udi  maximum  or  a  minimum  ;  and  as  we  wish  to  compare 
only  this  primitive  with  its  derived  curves,  we  must  consider 
^0,  //j,  etc.,  as  referring  to  this  primitive  and  to  the  given  limits 
only.  These  quantities  can  therefore,  so  soon  as  the  equation 
of  the  curve  and  the  values  of  x^  and  x^  are  known,  be  found. 
Hence  if  h^,  h^,  i^,  Zj,  etc.,  do  not  severally  vanish,  we  can 
make  L  assume  any  infinitesimal  value  we  please  by  suitably 
choosing  dy^^  dy^,  Sy^\  dy^\  etc.  But  if  the  solution  y  =z  f[x) 
cause  these  quantities  to  severally  vanish,  L  must  become  zero 
also,  and  no  other  condition  will  cause  L  to  vanish  necessarily 
without  restricting  the  values  of  Sy^^  6y^^  dy^',  etc. 


THEORE  TICAL  CONSIDER  A  TIONS.  4I 

38.  Let  us  now  consider  the  second  term, 


£y'y 


dx. 


In  this  integral  Sy  is  wholly  in  our  power,  being  subject  only 
to  the  condition  that  neither  it  nor  any  of  its  differential  co- 
efficients, to  the  7tth  inclusive,  shall  become  appreciable  within 
the  range  of  integration,  y"^^  being  the  highest  differential  co- 
efficient in  V.  In  other  words,  Sy  may  be  any  arbitrary  func- 
tion of  X  which  fulfils  these  conditions,  or  it  need  not  even  be 
the  same  function  throughout  the  entire  range  of  integration. 
On  the  other  hand,  M  is  not  in  our  power,  but  will,  as  in  the 
case  of  h^,  h^,  etc.,  depend  upon  the  equation  j  ^=zf{x).  Hence 
if  M  be  not  necessarily  zero  throughout  the  given  limits  of 
integration,  the  integral  will  be  wholly  in  our  power,  and  we 
may,  by  suitably  varying  y,  make  it  assume  any  infinitesimal 
value  we  please.  But  if  the  solution  y  =  f{x)  reduce  M  to 
zero  throughout  U,  then  the  integral  itself,  being  definite, 
must  become  zero ;  and  it  will  not  necessarily  vanish  under 
any  other  condition,  so  long  as  6y  is  wholly  unrestricted. 

39.  Resuming  equation  (8),  Ave  have 

L=-S.ySydx.  (9) 

Now  if  the  solution  j  =/(;i;)  be  such  as  to  cause  the  quantities 
/^o,  h^,  i^,  /„  etc.,  and  also  M  to  severally  vanish,  then  each 
member  of  (9)  will  likewise  vanish,  and  no  difficulty  will  occur. 
But  if  the  proposed  solution  be  not  able  to  fulfil  all  these  con- 
ditions, (9)  becomes  an  impossible  equation.     For  inasmuch 

as  L  and  J^    M  dydx  are  no  longer  necessarily  zero,  it  would 

in  effect  imply,  as  Prof.  Jellett  has  remarked,  "  that  the  inte- 
gral of  an  arbitrary  function  may  be  expressed  (without  deter- 


42  CALCULUS  OF   VARIATIONS. 

mining  or  even  restricting  its  general  form)  in  terms  of  the 
limiting  values  of  itself  and  a  certain  number  of  its  differen- 
tial coefficients.     This  is  manifestly  untrue." 

We  conclude,  then,  that  it  is  necessary  to  the  existence  of 
a  maximum  or  minimum  not  only  that  L  and  M  shall  vanish, 
but  that  each  of  the  quantities  h^,  h^,  z^,  z^,  etc.,  and  M,  shall 
become  zero. 

40.  Although  the  truth  of  the  preceding  principles  would 
appear  to  be  sufficiently  evident,  yet  Strauch,  one  of  the  most 
elaborate  writers  on  the  calculus  of  variations,  asserts  that  it 

cannot  be  proved  that  L  and  /     M  dy  dx  must  severally  vanish ; 

and  as  this  is  a  point  of  the  highest  importance,  and  of  some 
difficulty,  we  have  given  it  more  attention  than  it  has  generally 
received  hitherto.  Strauch  is,  however,  compelled  to  admit 
that  we  do  obtain  correct  results  by  this  method ;  and  there 
can,  as  Prof.  Todhunter  states,  be  no  doubt  that  the  principle 
is  sound. 

4-1.  Before  proceeding  further  we  will  apply  the  foregoing 
theory  to  the  solution  of  some  examples. 

Problem  IV. 

Let  V  be  any  function  of  y"  and  constants  only,  and  let  it  be 
required  to  detennijie  the  relations  which  vizist  szibsist  between  x 
and  y  in  order  to  maximize  or  minimize  the  expression 


u=syd^^ 


x^  and  x^  only  being  fixed. 
We  have 


SU=^  r^^^-^r  h"dx  =  r^Q  S/dx  =  o. 


THEORY  ILLUSTRATED.  43 

Then  transforming  dU,  as  just  explained,  and  denoting  by 
accents  total  differentials,  we  have 


+ir' 


Q'Sydx^Q,  (i) 


Whence,  since  M  must  vanish,  we  have 

Q'^o,        Q  =  c,        Q  =  cx  +  c'^  (2) 

If  we  had  supposed  the  values  y  and  y'  at  the  limits  to  be 
given  as  in  former  examples,  the  solution  could  be  carried  no 
further  without  determining  the  form  of  V.  But  since  dy^, 
^y^  ^Jo^  ^yl  ^^^  ^^^  necessarily  zero,  we  must,  from  the  pre- 
ceding discussion,  have  the  coefficients  of  these  quantities 
severally  zero.  Hence  2/  =  o,  QJ  =  o,  Q,  =  o,  Q^  =  o.  From 
the  third  and  fourth  of  these  equations,  combined  with  (2), 
we  have  cx^-{-  c'  =  o,  ex,  -\-  c'  =  o,  c{x^  —  x^  =  o.  Whence 
c  =  o,  and  then  c/  =  o.  Therefore  the  last  of  equations  (2) 
gives  Q  =  o. 

If  this  equation  is  to  hold  throughout  U,  y"  must  be  con- 
stant, although  it  may  have  several  constant  values.  Let  a 
be  one  of  the  roots  of  the  equation  Q  —  o.  Then,  as  y"  =  ^,  by 
integration  we  obtain 

j/=^  +  ^^  +  -5',  (3) 

the  equation  of  a  parabola ;  or  of  a  straight  line  if  a  should 
happen  to  become  zero. 

The  constants  b  and  b'  cannot  be  determined  so  long  as  the 
values  of  j^  and  y^  are  not  fixed.  For  it  is  easy  to  see  that 
the  equations  Q'  =  o  and  QJ  =  o  furnish  no  new  equations 


44  CALCULUS  OF    VARIATIONS. 

of  condition,  because  they  follow  from  Q  —  o^  and  any  values 
of  b  and  b'  which  satisfy  the  latter  will  also  satisfy  the  former 
two. 

Owing  to  its  simplicity,  we  may  also  examine  the  term  of 
the  second  order^  which  is 


2«^^o  dy 

Since  y"  is  a  constant,  ~^,  which  is  some  function  of  y\ 

dy 

must  be  also  a  constant,  say  A  ;  then,  since  the  terms  of  the 
first  order  vanish,  we  may  write 


2  ^^ 

which  shows  that  we  have  a  maximum  or  minimum  accord- 
ing as  A  is  negative  or  positive. 

Problem  V. 

42.  //  is  required  to  maxiinize  or  minimize  the  expression 

the  limiting  values  of  x  only  being  given. 
We  have 

SU=  r\^y"^y"  -  2^J)  dx  =  -  2yl"  Sy^  +  2y:"  dy^ 

+  2J'."  &J'/  -  2y:  5y:  +fj\2/-'  -  2)  6ydx  =  o.        (i) 


THEORY  ILLUSTRATED.  45 

Whence  equating  M  to  zero,  and  integrating,  we  have 


y'=i, 

(2) 

y^x  +  a. 

(3) 

y^'^-^ax  +  b, 

(4) 

y^^  +  '^  +  ^^+c, 

(5) 

X'*    .   ax^   \    i>x^    ,          ,     , 
^  =  24+6    +2    +^"  +  '^- 

(6) 

Now  if  dy^  and  Sy\  be  unrestricted,  we  must  have,  from  (i), 
y^"  —  o,y^"  —  o,  which  give,  in  (3),  x\-\-a=  o,  and  x^-\-a-=o, 
which  are  impossible  equations,  since  x^  and  x^  are  not  to  be 
equal.  Whence  we  conclude  that  the  solution  will  not  be 
possible  unless  we  restrict  Sy^  and  Sy^,  so  that  y^"  and  y^" 
need  not  severally  vanish. 

We  will  now  suppose  y^  and  j,  to  be  given,  but  j/  and  j/ 
to  be  unrestricted.  Then,  from  (i),  we  must  have  y"  =  o, 
y^'  =  o,  which  would  give,  in  (4), 


^^ax,  +  d  =  o,  (7) 


x: 


2 


-J^ax,  +  d  =  o,  (8) 


From  these  equations  we  readily  obtain 


^  =  -^-(^o+^^)y  (9) 


2 


i  =  ^.  (10) 


46  CALCULUS  OF   VARIATIONS. 

Now  suppose,  for  simplicity,  that  we  take  x^  equal  to  any 
constant  e,  and  x^  to  —  e.     Then   (9)  and  (10)  give  ^  =  o, 

b  = ,  and  (6)  would  become 

x"      e'x'' 

y  = \-cx-\-d.  (11) 

24         4     '  '  ^     ^ 

But  it  will  be  remembered  that  this  equation  is  only 
admissible  on  the  supposition  that  we  are  able  to  make 
h^^y^  —  hfy^  vanish ;  and  as  h^  and  h^  cannot  severally  vanish, 
this  is  accomplished  by  fixing  the  values  of  y^  and  y^,  and  the 
assignment  of  these  values  will  afford  us  the  conditions  for 
determining  the  remaining  constants.  Equation  (11)  now 
gives 


-^f+ 

ce  +  d, 

(12) 

'■---^- 

ce^d. 

(13) 

Whence  we  obtain 

,_yr-yo 

'-      2e     ' 

(14) 

"=-+£ 

2 

(15) 

Suppose,  for  still  greater  simplicity,  we   take   the  fixed 
points   on   the   axis   of  x.     Then   (14)   and    (15)  give  c  =  o, 

d  =  ^—,  and  we  shall  have,  finally, 

24 

_x'  _e^\5/ 
-^""24        4         24* 

But  suppose,  as  usual,  the  limiting  values  of  y  and  y  were 
both  given,  and  let  us  consider  the  particular  case  m  which 


THEORY  ILLUSTRATED.  A,7 

we  have  x^  =  e,  x,=^  —  e,  y,  =  o,  y^  =  o,  y/  =  i,  yj  =  —  i. 
Then,  from  (5),  we  have 


Then  eliminating  c  between  (17)  and  (18),  we  have 
and  from  the  same  equations  we  obtain 

^  =  -J-'  (20) 

Moreover,  substituting  in  turn  e  and  —  e  for  x  in  (6),  we  have 

■^■  =  ^  +  T  +  T  +  ^'  +  '^'  (^') 

Eliminating  dy  we  have 

—  +  2^^  =  o.  (23) 

Substituting  for  a  its  value  from  (20),  we  find  ^  =  o,  whence 
also  a  =  0;  and  again  substituting  these  values  with  that  of  b, 
(21)  gives 

24        2 


48  CALCULUS  OF   VARIATIONS. 

Now  substituting  these  values  in  (6),  we  have,  finally, 

y— x"  -\ . 

24  \2e  24      2 

The  term  of  the  second  order  is  merely 

which  is  of  coUrse  positive,  thus  giving  a  minimum.  That  is, 
any  solution  which  reduces  the  terms  of  the  first  order  to 
zero  will  render  U  a  minimum.^ 

43.  Now  resume  for  a  moment  the  consideration  of  Prob. 
I.     There  we  have 

which  give  y^  =  o,  and  7/  =  o.  But  since  we  know  from  the 
general  solution  that  y'  —  a,  these  two  conditions  are  in  reality 
only  one,  a  =^  o.  Hence  if  no  restrictions  be  imposed  except 
that  x^  and  x^  shall  be  fixed,  the  line  must  be  parallel  to  the 
axis  of  X, 

But  the  constant  b  cannot  be  determined  in  this  case.  In- 
deed it  is  evident  that  the  straight  line  parallel  to  x  is  shorter 
than  any  other  curve,  or  straight  line  even,  which  can  be  drawn 
having  x^  and  x^  as  the  abscissae  of  its  extremities,  and  that 
hence  our  first  result  is  confirmed.  Moreover,  since  the 
length  of  this  line  will  be  the  same,  whatever  be  its  distance 
from  the  axis  of  x,  the  value  of  b  can  have  no  effect  upon  its 
length,  and  therefore  ought  to  remain  undetermined.  If, 
however,  the  co-ordinates  of  one  of  its  extremities  be  given, 
the  line  becomes  a  parallel  to  x  through  that  fixed  point,  and 
b  is  determined. 

*  The  last  two  examples  are  from  the  Adams  Essay,  by  Prof.  Todhunter  (p.  15), 
but  have  been  considerably  elaborated. 


THEORY  ILLUSTRATED.  49 

4-4.  Next  consider  Prob.  II.,  Case  i.     There  we  find 

But  from  equation  (8),  Art.  17,  if  we  make  h^  or  h^  zero^  we 
see  that  the  equation 


y 


—  Q 


must  hold  throughout  the  curve,  and  this  gives  y'  =  o,  which, 
as  it  denotes  the  vertical,  is  the  true  solution.  For  if  a  par- 
ticle be  merely  required  to  descend  from  one  horizontal  plane 
to  another,  it  will  do  so  along  the  vertical  sooner  than  along 
any  other  line.  The  equation  of  this  vertical  is  y  =  d,  in 
which  the  value  of  d  can  have  no  effect  upon  the  time  of 
descent,  and  therefore  remains  undetermined,  as  it  should. 

Next  consider  the  second  case  of  the  same  problem'.    There 
we  have 


The  first  of  these  equations  gives j//  =  o ;  and  since  V{i  -\-y''')y 
—  V2r,  r  being  the  radius  of  the  generating  circle,  we  have 

.(  |/(i+y>f  „       V2r 

and  this,  if  equated  to  zero,  will  give  //  =  o,  which  is  evi- 
dently impossible.  Hence  h^  cannot  be  zero  ;  and  to  make  the 
term  h^Sy^  vanish,  we  must  assign  the  value  oi  y^. 

Now  it  will  be  remembered  that  the  general  solution  was 
a  cycloid,  having  a  cusp  at  the  starting-point  of  the  particle, 
and  that  b  was  merely  the  value  oi  y^,  which  is  now  deter- 
mined. Moreover,  since  we  have  just  found  that  the  tangent 
to  this  cycloid  at  the  point  which  is  not  fixed  must  be  par- 


50  CALCULUS  OF    VARLATLONS. 

allel  to  the  axis  of  x,  it  follows  that  its  vertex  must  be  at  this 
point.  Hence  the  generating  circle  must  be  such  that  it 
would  roll  through  a  semicircle  while  its  centre  was  de- 
scribing the  distance  x^—  x^,  and  therefore  we  have 

^1   —  ^0 

r  =  ~ -. 

45.  Let  us  in  the  last  place  consider  Prob.  III.  If  we 
could  have  fully  integrated  equation  (lo),  Art.  30,  the  inte- 
gral would  have  involved  four  constants,  and  for  determining 
these  constants  we  would  have  7/,  y^,  j/^,  y^  equal  to  four 
assigned  quantities.  It  would,  however,  be  too  tedious  to 
discuss  this  case  in  detail,  and  we  will  next  suppose  the  values 
of  7o  and  y^  to  be  fixed,  while  those  of  j//  and  7/  are  variable. 
Then  equating  to  zero  the  coefficients  of  ^yl  and  Sy^^  we 
shall  have 


^l.= 


and  since  \/  i-\-y''^  cannot  be  zero,  y"  and  y"  must  each  be 
infinite,  thus  giving  the  cycloid  cusps  at  the  two  fixed  points. 
Let  b  denote  the  angle  which  the  line  joining  these  cusps 
makes  with  the  axis  of  x.  Then  b  is  identical  with  b  of  equa- 
tion (16),  Art  30,  and  is  at  once  determined,  its  tangent  being 

y^  -  y. 


■^1-^0 


Then,  also, 


8  27t 


Let  us  now  suppose  the  values  of  y^  and  y^  to  be  unre- 
stricted. Then  we  must  equate  the  coefficients  of  ^y^  and 
djo  severally  to  zero,  which  will  give  the  equation 


THEORY  ILLUSTRATED.  51 

and.  a  similar  equation  for  the  lower  limit.  But  from  equa- 
tion (it),  Art.  30,  the  first  member  of  the  last  equation  equals 
—  c,  making  c  in  this  case  zero.  Therefore  equation  (13)  of 
the  same  article  becomes 

(i+yy_    d 

y"  2* 

Now  d  cannot  vanish.     For  if  it  can,  we  must  either  have 


\/\  4-y  =  o,  which  would  render  y  imaginary,  ox  y"  must  be 
infinite  throughout  the  curve,  which  is  also  inadmissible.  But 
if  dy^  and  Sy^  do  not  vanish,  we  must,  as  we  have  just  seen, 
have  y"  and  y^'  infinite.  It  follows,  therefore,  that  j//  and  7/ 
must  become  infinite,  as  d  would  otherwise  vanish. 

We  conclude,  then,  that  the  cycloid  must  in  this  case  be 
so  placed  as  to  have  the  line  joining  its  cusps  parallel  to  the 
axis  of  X.     Then  we  shall  evidently  have 


^  ^  ^,  -  ;r. 


27r 

while  the  constant  angle  b  of  equation  (16),  Art.  30,  will  be- 
come zero,  and  it  is  easy  to  show  also  that  <^  =  8r  =  h. 

46.  It  is  evident  that  none  of  the  results  of  the  preceding 
articles  could  be  confirmed  as  maxima  or  minima  without  an 
examination  of  the  sign  of  the  terms  of  the  second  order, 
because  even  if  those  terms  were  shown  to  be  certainly  posi- 
tive or  negative,  in  any  particular  problem,  by  making  any 
of  the  variations  Sy^,  Sy^^  6y^',  dy^',  etc.,  zero,  it  would  not  fol- 
low that  we  could  be  certain  of  the  same  sign  when  those 
restrictions  were  removed  or  modified. 

But  it  will  be  remembered  that  in  the  problems  thus  far 
discussed  we  have,  with  the  exception  of  Case  2,  Prob.  II., 
been  able  to  determine  the  sign  of  the  terms  of  the  second 
order  without  imposing  any  restriction  upon  the  variations 
of  y  and  y'  at  the  limits.     The  only  result,  then,  which  we 


52  CALCULUS  OF    VARIATIONS. 

have  to  confirm  is  this :  that  when  the  starting-point  of  the 
particle  is  given,  its  terminal  point  being  restricted  to  have  a 
given  abscissa  x^,  the  curve  of  quickest  passage  from  x^  to  x^ 
will  be  a  cycloid  with  a  cusp  at  the  first  point,  and  its  vertex 
at  the  second.  An  examination  of  equations  (12)  and  (13), 
Art.  27,  will  show  that  if  we  had  not  supposed  Sy^  and  dy^  to 
be  zero,  equation  (19)  of  the  same  article  would  have  become 


27 

in  which  the  integral  is  positive  as  before.  But  by  hypothesis 
dj/„  =  o,  and  asy  vanishes  at  the  vertex,  while  j^  becomes  a  or 

2r,  we  have  (— )  =  o.     Hence  both  terms  without  the  sign  of 

integration  vanish,  and  we  have  a  minimum  as  before. 

4-7.  We  may  now  proceed  without  difficulty  to  that  gen- 
eral discussion  of  the  terms  of  the  first  order  which  is  usually, 
but  unadvisedly  we  think,  presented  prior  to  the  discussion 
of  particular  problems. 

Assume  the  equation  U  —    I      Vdx,  where  V  is  any  func- 

tion  of  X,  y,  y'  .  .  .  .  y^\  and  let  it  be  required  to  determine 
what  function  y  must  be  of  x  in  order  to  render  U  a  maxi- 
mum or  minimum.  Then  finding  dU,  and  transforming  it  by 
integration  as  far  as  possible,  and  then  equating  to  zero  sever- 
ally the  coefficients  of  dy^,  dy^,  etc.,  together  with  M,  which  is 
the  coefficient  of  Sy  dx  under  the  integral  sign,  we  obtain  the 
equations  h^  =  o,  h^  —  o,  i,  =  o,  z^  =  o,  etc.,  and  also  M  =  o, 
where,  as  will  be  remembered, 

,,      dP    ,    d'Q 


THEORETICAL   CONSIDERATIONS  RESUMED.  53 

all  the  differentials  being  total,  and  iV,  P,  Q,  etc.,  being  the 
partial  differential  coefficients  of  V  with  respect  to  y,  y\  y", 
etc. 

Now  the  equation  M  =  o  will,  in  general,  be  a  differential 
equation  of  the  order  2;/,  because  its  last  term  will  be 

d^     dV 


which  will  usually  involve 


dx' 


=  (yw))(«)  =  y 


{2ny 


Hence  the  complete  integral  of  this  equation  must  usually 
contain  27t  arbitrary  constants,  and  may  be  supposed  to  be 
put  under  the  form 

y  =  f{x,  C„  C„ c,n)  =  /.  (i) 

Now  since  every  solution  of  our  problem  must  satisfy 
the  equation  M=o,  it  must  also  be  comprised  in  (i),  which 
establishes  a  general  relation  between  x  and  y,  or,  in  other 
words,  gives  us  some  plane  curve  ;  which  relation  or  curve  is, 
however,  capable  of  great  modification,  by  adjusting  suitably 
the  values  of  these  2n  arbitrary  constants. 

48.  If  now  we  examine  the  equations  //,  =  0,  ^0  =  o>  ^tc, 
which  we  may  call  the  equations  at  the  limits,  we  shall  find 
that  their  number  is  also  271.  Moreover,  these  equations,  not 
holding  throughout  the  curve,  do  not  establish  any  general 
relation  between  x  and  y,  as  did  the  equation  M=o,  but 
merely  fix  the  conditions  which  the  required  curve  must  fulfil 
at  the  limits.  This  is  as  it  should  be.  For  if  the  equations 
h  =  o,  i  —  o,  etc.,  could  be  supposed  to  hold  throughout  the 
curve,  they  would  each  establish  a  relation  between  x  and  y, 
and  unless  these  relations  should  happen  to  agree  with  each 


54  CALCULUS  OF    VARLATLONS. 

Other,  and  also  with  that  derived  from  the  equation  M  =  o, 
which  would  seldom  if  ever  occur,  the  solution  would  become 
nugatory. 

Now  suppose  the  complete  integral  of  the  equation  M  =o 
were  obtained,  and  expressed  as  in  (i).  Then  if  the  form  of/ 
were  known,  we  could  form  the  expressions  h^,  h^,  t^,  etc.,  and 
these  expressions  would  all  be  known  functions  of  either  x^  or 
x^,  together  with  some  of  the  2n  arbitrary  constants,  no  vari- 
able entering  these  functions,  because  x^  and  x^,  being  assigned 
quantities,  may  be  regarded  as  constants  also. 

We  see  then  that  in  the  equations  h^  =  o,  k^  =  o,  etc.,  we 
have  2n  equations  between  x^  and  x^  which  are  assigned,  and 
2?t  arbitrary  constants,  and  should  therefore  be  able  to  deter- 
mine these  2n  constants  in  terms  of  the  known  constants  x^ 
and  x^. 

Now  suppose  the  limiting  values  of  y^  and  y^  were  given. 
Then,  since  the  variations  of  these  quantities  would  become 
zero,  /^,  and  /i^  would  no  longer  necessarily  vanish.  But  in 
this  case  it  is  evident  that  the  two  equations  thus  lost  would 
be  replaced  by  the  equations  jk^  —  f{x^,  c^,  c^  .  .  .  .  c^n)  —fv  ^nd 
y.  =  A^o,  c,,c^ c^y^  =  /„ ;  and  as  jk^  and  y,  are  now  sup- 
posed to  have  assigned  values,  the  number  of  the  equations 
for  the  determination  of  the  2n  constants  remains,  as  before,  2;/. 
In  like  manner,  if  ^//  and  Sj/J  should  become  zero,  the  con- 
ditions i,  =  o  and  t,  =  o  would  disappear.  But  to  supply 
their  place  we  would  have  the  equation 

and  a  similar  equation  for  the  lower  limit,  j//  and  j/  being 
now  assigned  constants  also ;  so  that  we  still  have,  as  before, 
2n  ancillary  equations. 

Suppose,  lastly,  that  any  of  the  variations  Sy^  ^y^,  Sy^',  etc., 
were  connected  by  given  equations,  and  suppose  there  were 
m  such  equations.     Then  if  we  should  express  as  many  of  the 


EXCEPTIONS    TO    THEORY.  55 

variations  as  possible  in  terms  of  the  remaining  variations, 
and  then  equate  to  zero  the  coefficients  of  the  several  varia- 
tions in  the  reduced  system,  it  is  plain  that  our  ancillary 
equations  would  be  only  2n  —  m  in  number.  But  since  we 
have  the  m  equations  between  certain  variations,  we  are  evi- 
dently able  to  form  new  systems  of  independent  variations  in ' 
such  a  manner  as  to  obtain  7n  more  equations  between  x^,  x^, 
and  the  2n  constants. 

Thus  we  see  that,  theoretically  at  least,  the  terms  at  the 
limits  furnish  us  with  2n  equations  for  the  determination  of 
the  2n  arbitrary  constants,  which  would  in  general  occur  in 
the  complete  integral  of  the  equation  M—o,  and  that  what- 
ever condition  reduces  the  number  of  the  original  equations, 
by  annulling  or  combining  two  or  more  of  them,  will  at  the 
same  time  furnish  in  their  place  as  many  new  equations  for 
the  determination  of  these  constants  as  have  been  removed. 

49.  The  preceding  considerations,  which  are  theoretical, 
require  some  modification,  first  as  regards  the  terms  at  the 
limits,  and  second  as  regards  the  equation  M  =  o.  With 
regard  to  the  terms  at  the  limits,  it  has  probably  been  noticed 
that  it  has  not  been  in  general  possible  to  satisfy  all  the  equa- 
tions //,  =  o,  //„  =  o,  etc.,  as  some  of  these  equations  become 
conflicting.  But  even  in  these  cases  we  can,  as  we  have  seen, 
generally  obtain  2n  harmonious  equations  by  restricting  one 
or  more  of  the  variations ;  as,  for  example,  by  supposing  dy^, 
dj/^,  or  (^K/,  etc.,  to  vanish. 

In  fact,  the  occurrence  of  these  conflicting  equations  de- 
notes merely  that  the  problem  in  its  present  form  is  not 
capable  of  solution,  and  as  it  might  be  foreseen  that  such 
questions  would  present  themselves,  the  occurrence  of  these 
conflicting  equations  would  naturally  be  expected. 

50.  The  following  exceptions  may  be  regarded  as  due  to 
the  nkture  of  the  equation  M  =  o,  although  they  properly 
arise  from  the  nature  of  the  function  V. 


5^  CALCULUS  OF    VARLATIONS. 

Exception  i.  Suppose  N  to  vanish  in  the  equation  M  =  o, 
which  would  of  course  happen  if  y  did  not  exphcitly  enter  V. 
Then  we  would  have 


whence 


dx^  dx'       ^^'^•-O' 


ax         dx 


But  the  first  member  of  the  last  equation  equals  h ;  and  as  h 
must  vanish  at  either  limit  unless  the  values  of  )\  and  y^  be 
assigned,  we  have  c  =.o\  and  since  the  equations  h^  =  o  and 
//o  =  o  are  each  satisfied  by  this  value  of  c,  they  furnish  no  new 
condition  for  the  determination  of  any  other  constant  which 
may  enter  the  complete  integral  of  the  equation  M  =  o.  Thus 
the  conditions  furnished  by  the  terms  at  the  limits  are  in  this 
case  reduced  to  2;^  —  i,  two  of  them  having  become  identical. 
If,  however,  the  value  of  either  y^  or  y^  be  assigned,  this  will 
furnish  a  new  equation  of  condition  which  will  compensate 
for  that  which  was  lost. 

This  case  is  fully  exemplified  by  the  discussion  of  Prob.  I. 
in  Art.  43  and  Prob.  II.,  Case  i,  in  Art.  44. 

Similarly,  suppose  V  to  contain  neither  y  nor  y'.  Then 
we  would  have 

,^      d'Q      d'R    ,     ^  ^  '       ^ 

dO      d'R 


dx        dx^ 


etc.  =  a,  (2) 


G-g  +  etc.  =  «^  +  ^.  (3) 

Now  if  the  limiting  values  of  y  are  variable,  we  have  h^  =  o 
and  //„  =  o ;  and  it  is  easy  to  see  that  in  this  case,  as  P  is  want- 


EXCEPTIONS   TO    THEORY.  $7 

ing,  the  first  member  of  (3)  is  z,  and  that  of  (2)  is  —  A,  and 
therefore  we  have  ax^  -J-  <$>  =  o,  and  ax^  -f-  ^  =  o,  whence  we 
find  a  —  o  and  d  =  o,  and  (3)  becomes 

Now  it  must  be  remembered  that  this  equation  has  been  de- 
duced solely  from  the  conditions  z,  —  o  and  i^  =  o.  But  dif- 
ferentiating (4),  we  have 

dQ      d'R    ,     ^  J 

-^ -^  +  etc.  =  0,        QT    —h=  o. 

dx        dx 

Whence  it  appears  that  since  the  equations  h^  =  o,  //„  =  o,  can, 
without  involving  any  other  relations,  be  deduced  from  the 
equations  z^  =  o,  z^  =  o,  they  furnish  no  new  data  for  the  de- 
termination of  the  constants  which  will  be  found  in  the  com- 
plete integral  of  the  equation  M  =  o.  Hence  in  this  case  our 
ancillary  equations  will  furnish  but  271  —  2  distinct  conditions, 
thus  leaving  generally  two  constants  undetermined,  unless 
one  or  more  additional  equations  be  supplied  by  assigning 
the  values  of  one  or  more  of  the  quantities  j/j,  jo,  j/,  y^\  etc. 
This  case  is  fully  exemplified  in  the  discussion  of  Prob.  IV. 

Generally,  if  the  first  in  of  the  quantities  y,  y',  y",  etc.,  be 
wanting  in  F",  while  at  the  same  time  the  variations  of  these 
quantities  at  the  limits  remain  unrestricted,  in  arbitrary  con- 
stants in  the  general  solution  must  also  remain  undetermined. 

51.  Exception  2.  Suppose  V  to  contain  only  the  first 
power  of  y^^  the  highest  differential  coefficient  which  is  in- 
volved. Then  in  this  case  the  equation  M=o  cannot  be  of 
an  order  higher  than  2n  —  i.    For  the  last  term  in  M  must  be 

±  -z-^  ~rj^y  ^^^  ^^  ^"^^  *^^  ^^^^  power  of  y^)  occurs  in  V,  the 


58  CALCULUS  OF    VARIATIONS. 

partial  differential  coefficient  of  V  with  respect  to  y*^>  will  not 
contain  that  quantity  at  all.  Whence  it  is  evident  that  M 
cannot  be  of  an  order  2n ;  and  indeed  Prof.  Jellett  has  shown 
that  it  cannot  in  this  case  rise  above  the  order  2n  —  2  (see  his 
page  46),  but  it  does  not  seem  necessary  to  reproduce  his 
proof  here. 

Now  in  this  case  the  equations  at  the  limits  will  be,  as 
before,  2n  in  number,  while  the  constants  in  the  complete  in- 
tegral of  the  equation  M  —o  will  not  exceed  in  number  2n—  \, 
and  in  fact  will  not  exceed  2n  —  2.  This  seeming  exception 
is,  however,  explained  by  the  fact  that  in  all  such  cases  the 

integral  U^  or  J  Vdx,  is  capable  of  being  reduced  by  integra- 
tion  to  the  form  ^==/i  — /o+  /     V'dx^  where  f  and  /,  are 

t/  Xq 

quantities  free  from  the  sign  of  integration,  while  V^  does  not 
contain  any  differential  coefficient  of  y  higher  than  y^  -  ^) ;  and 
we  will  next  show  that  this  reduction  can  be  effected. 

62.  Let  y^-)  be  the  highest  differential  coefficient  in  V. 
Then,  since  its  first  power  only  occurs  in  F,  we  may  write 

j7_^yn)_|_^^  (l) 

where  w  is  that  part  of  F  which  is  a  factor  of  y*^\  and  z  the 
other  terms  of  V,  both  being  of  course  of  a  lower  order  than 

y^^\     Then  the  equation  U  —  J^'  Vdx  becomes 

nx-^  nxi 

U—J     wy^'^^dx-^J     zdx.  (2) 

But  we  are  evidently  able  to  form  the  following  equation : 

fwy^^)dx  =  W+fzdx,  (3) 


EXCEPTIONS   TO    THEORY,  59 

where  Wand  Z  are  functions  at  present  unknown.  For  this 
equation  can,  if  in  no  other  manner,  always  be  formed  thus : 

/  wy^'^^dx  —  wy^'^^x  -\-J  —  — -  zvy^'^'^.xdx.  (4) 

But  W  and  Z  can  be  so  taken  that  the  second  member  of  (3) 
will  contain  no  higher  differential  coefficient  than  y^--^),  be- 
cause (3)  can,  in  the  following  manner,  be  satisfied  upon  this 
assumption.     First  differentiate  (3),  and  we  shall  have 

,,       ^  ,  dW  ,  dW  ,  ,  dW  „  ,     ^      ,     dW      ,,         ,  , 

^yin^  =  Z-Y^^+^^-y+-^^ry  +  ^tC.  +  ---/n,.  (5) 

which  must  be  the  complete  differential  of  (3)  if  our  assump- 
tion be  true,  but  not  otherwise.  But  (5),  and  consequently  (3), 
will  be  satisfied  if  we  put 

„      dW  ^  dW  ,   ^     ^      ,      dW      ,     ,. 

-  ^  =  ^-  +  -dy-y  +  ^^"  +  ^y^)-^'^"  •  W 

Therefore  ^is  found  by  integrating  w  with  respect  to  y^^-i) 
only.     Hence,  finally,  we  have* 

=  W,-W,+  r'v'dx.  (8) 

This  case,  then,  is  in  reality  no  exception  at  all,  because  the 


*  This  theorem  is  due  to  the  great  Euler  (see  Meth.  Inven.,  pp.  62,  75),  and 
has  been  nearly  reproduced  by  Prof.  Jellett  on  his  page  46. 


6o  CALCULUS  OF  VARIATLONS. 

difficulty  arises  merely  from  the  fact  that  the  original  integral 
had  not  been  reduced  to  its  lowest  terms.  For  although  we 
have  not  yet  considered  the  class  of  problems  to  which  this 
reduced  form  of  U  belongs,  it  is  easy  to  see  that  the  equation 
M  =  o,  resulting  from  V^  only,  will  not  now  be  of  an  order  ex- 
ceeding 2n  —  2,  which  is  the  result  obtained  by  Prof.  Jellett. 

53.  Exception  3.  Let  V  be  of  the  form  y/-\-F,  where  / 
contains  only  quantities  incapable  of  variation,  e.g.  x  and  con- 
stants, and  F  may  contain  any  quantities  except  j/.  Then 
JV  becomes  simply  /,  and  the  equation  M  =  0  will  give  the 
equations 


dx       dx^ 


etc.  =r/. 


Now  the  first  member  of  (i)  equals  ]i\  and  if  we  suppose 
J/,  and  /o  to  be  unrestricted,  we  must  have  h^  —o,  i^  =  o ;  and 
using  these  restrictions,  (i)  will  give 

{/WJ  +^  =  0,  (2) 

and 

'■/Or)}  +.  =  0.  (3) 

But  as  the  first  members  of  (2)  and  (3)  contain  only  one 
indeterminate  constant,  c,  it  will  in  general  be  impossible  to 
satisfy  both  equations,  and  the  problem  in  this  form  does  not 
usually  admit  of  a  solution.  But  if  we  make  /  zero,  so  that 
V  is  any  function  not  containing  /,  the  problem  becomes  a 
case  of  Exception  i,  and  may  or  may  not,  according  to  its 
nature,  be  capable  of  a  general  solution,  one  constant  at  least 


GENERAL  FORMULA,  6 1 

remaining  undetermined.  This  exception  is  exemplified  by 
Prob.  v.,  in  which /=  -  2,  F  =  y"\ 

64.  It  is  now  evident  that  if  we  require  that  U  shall  be  a 
maximum  or  minimum,  the  calculus  of  variations  will  ter- 
minate its  aid  in  the  discussion  by  leaving  us  with  a  series  of 
differential  equations,  that  of  the  highest  order  holding  true 
for  all  values  of  x  from  x^  to  x^,  the  others  merely  holding  at 
the  limits  of  integration.  From  the  former  of  these  equations, 
as  it  is  general,  the  general  solution  must  be  obtained,  and 
then  the  remaining  or  ancillary  equations,  not  being  general, 
mpst  be  satisfied,  if  they  can  be  satisfied  at  all,  by  the  assign- 
ment of  suitable  values  to  the  constants  which  will  occur  in 
the  general  solution ;  or  we  may  say  that  these  ancillary 
equations  determine  the  values  of  the  constants. 

The  determination  of  these  constants  is  not  in  general  dif- 
ficult when  the  complete  integral  of  the  equation  is  known ; 
but  this  integral  is  often  obtained  with  difficulty,  and  is  some- 
times altogether  unobtainable.  In  fact,  this  difficulty  is  anal- 
ogous to  that  which  is  frequently  experienced  in  solving  the 
final  equation  or  equations  of  condition  given  by  the  differ- 
ential calculus  in  the  discussion  of  an  ordinary  problem  of 
maxima  or  minima,  except  that  in  the  former  case  the  final 
equations  are  differential  and  must  be  solved  by  the  calculus, 
while  in  the  latter  they  are  algebraic  and  must  be  solved  by 
the  theory  of  equations. 

55.  We  shall  next  proceed  to  establish  some  principles  re- 
garding the  integrability  of  the  equation  M  =  o,  and  to  deduce 
some  formulae  which  will  be  found  useful  in  our  subsequent 
discussions. 

Suppose,  in  the  first  place,  that  the  first  m  of  the  quanti- 
ties N,  P,  Q,  R,  etc.,  were  wanting  in  the  equation  M  =  o, 
which  would  of  course  happen  if  the  first  m  of  the  quantities 
J>  y'l  y"  1  y'" y  ^tc,  were  wanting  in  F;  then  the  equation 
J/=  o  can  be  integrated  at  least  m  times. 


62  CALCULUS  OF. VARIATIONS. 

For  let  m  be  4,  for  example.     Then  we  would  have 

which,  being  integrated  four  times,  becomes 

S V-  etc.  =  -—A \-  ex  -\-  d, 

dx  62 

and  similarly  if  m  were  any  other  number. 

56,  Suppose,  in  the  second  place,  that  the  independent 
variable  x  does  not  occur  explicitly  in  V\  then  the  equation 
M=  o  can  be  integrated  at  least  once.  For  since  V  does  not 
contain  x,  we  have 

dV=  Ndy  +  Pdy'  +  Qdy"  +  Rdy'"  +  etc. 

=  {Ny' -\-Py" -\-Qy"'^Rf'^^\.Q>,dx,  (i) 

Now  substituting  in  the  last  member  of  (i)  the  value  of  N 
derived  from  the  equation  M  —o,  viz., 

jyj_dP      d'Q     ^^^ 
dx       dx^ 

we  shall  have 

But    every  parenthesis   in   (2)    can   be   integrated  by  parts. 
Taking,  for  example,  the  third,  and  recollecting  that 

y  ^  dx  =  dy"\        y"'dx  =  dy\        y"dx  =  dy\ 


GENERAL  FORMULA.  ^3 


we  have 


Jr^^  dx  =  Ry'"  -J^^y["dx, 
rdR   ,„,  dR   „  .    rd'R    „, 

rd'R  „,      d'R  ,     rd'R  ,, 


Hence 


/{^/'  +  /^l^.=/^y"-fy'+^y.     (3) 

Integrating  the  remaining  terms  in  a  similar  manner,  we 
would  have 

which  equation  is  certainly  of  an  order  lower  than  that  of  the 
differential  equation  M  —  o. 

The  following  particular  cases  of  this  formula  are  given 
for  convenience  of  reference  : 

First.    If  F  be  a  function  of  y'  only,  we  shall  have,  from  (A), 

v=c  +  py.  (B) 

But  since  in  this  case  F  is  a  function  of  y,  P  must  also  be  a 
function  of  y' ;  so  that  (B)  may  be  written 

Ay')+/p{y)--=c  =/'{/), 

where  f  is  an  arbitrary  function.  The  last  equation  can 
therefore  only  be  satisfied  by  making  y  a  constant,  say  y'  =  r„ 
which  gives  y=CiX-\-c^. 


64  CALCULUS   OF  VARIATIONS. 

Hence  if  we  require  the  nature  of  the  curve  which  will 

maximize  or  minimize  the  expression  U  =^       ^Vdx,  where  V  \s> 

any  function  of  y'  only,  the  straight  line  is  the  solution,  if 
there  be  a  solution ;  that  question  being  decided  by  an  appeal 
to  the  terms  of  the  second  order. 

Second.  If  F  be  a  function  of  y  and  y'  only,  (A)  will  still 
give 

v=c+py.  (C) 

Third.  If  F  be  a  function  of  y  and  y'^  only,  then  (A)  will 
give 

F-.+  e/-gy.  (D) 

57.  Suppose,  in  the  third  place,  that  the  independent  vari- 
able Xy  and  also  the  first  in  of  the  quantities  y,  y',  y%  etc.,  are 
wanting  in  F;  then  the  equation  M  ^^  o  can  be  integrated  at 
least  m  -\-  i  times.  Let  in,  for  example,  be  4  as  formerly. 
Then  the  equation  M  =^  o,  after  having  been  integrated  four 
times,  according  to  the  first  case,  and  using  /,  q,  r,  s,  etc.,  for 
P,  Q,  R,  S,  etc.,  to  prevent  confusion,  becomes 

s — \-  — -  —  etc.  =  ax^  -\-  bx"  •\-  ex  -\-  d,  (i) 

dx       dx 

Also,  we  have  the  equation 

dV  ^sdf^  -^  tdf""^  +  iidy"^^^  +  etc. 

=  (^y^)  +  /y^)  +  uy^"^  +  ^y«^  +  etc.)  dx.  (2) 

Substituting  in  (2)  the  value  of  s  derived  from  (i),  we  have 

dv  =  (/y^>  +  ^  f\  dx  +  (^^y  *)  -  ^/  j'(^^)  dx 

-j-  / ^y ')  +  ^^ y5)j  ^^  _j_  etc.  +  {ax'  +  bx'  J^cx-\-d)  /'^  dx.    (3) 


GENERAL   FORMULA.  65 

Integrating  by  parts,  as  in  the  second  case,  we  have 

+f{ax'  -^  bx' -^  ex -\- d)  f  dx.  (4) 

Moreover,  the  integral  sign  can  easily  be  removed  from  the 
remaining  terms  in  (4).     For,  by  parts,  we  have 

/  ax^y^^^dx  —  ax""/^  —J  ^axY^'dx, 
-fiax^dx  =  -  saxy  +f6axy''dx, 
I  6axy"'dx  =  6axy"  —  I  6ay"dXy 

r 

—  J  6ay"dx  =  —  6ay\ 
Hence 

/  axy^^^dx  =  ax^f^  —  ^ax^y'"  +  6axy"  —  6ay' ; 

and  in  like  manner  we  may  integrate  all  the  other  terms. 

Thus,  for  example,  in  Prob.  IV.  we  find,  after  two  integra- 
tions of  M, 

Q     or     _,=.^  +  ., 

which,  being  again  integrated,  gives 

V=  cxy"  -  cy'  +  c'y"  +  d. 


65  CALCULUS   OF  VARIATIONS. 

Or,  let   V  be  a  function  of  y'  and  y"  only.      Then,  after  one 
integration  of  M^  we  have 

P f^  =  <^. 

ax 

We  also  have 

and  substituting  the  value  of  P  from  the  preceding  equation, 
we  have 

dV  =  [af  +  Qy'"  +  ^yy^y 
which,  being  integrated,  gives 

Problem  VI. 

68.  //  is  required  to  determine  the  form  of  the  solid  of  revo- 
lution which  will  experience  a  niinimiun  resistance  in  passing 
through  a  homogeneous  fluid  in  the  direction  of  x,  the  axis  of 
revolution  of  the  solid. 

Although  it  is  evident  that  the  problem  does  not  admit  of 
a  solution  until  some  further  restrictions  are  imposed,  we  shall 
at  present  merely  assume  that  the  distance  x^  —  x^  is  given. 

Let  ds  be  an  element  of  the  generating  curve,  pds  the  nor- 
mal pressure  which  it  experiences  in  passing  through  the  fluid, 
and  V  its  velocity  in  the  direction  of  that  normal,  or  the  velo- 
city with  which  the  particles  of  the  fluid  are  displaced  by  it 
in  that  direction.  Then,  adopting  the  usual  theory  regarding 
the  pressure  and  resistance  of  fluids,  we  have 

pds  =  czf'ds,  (i) 


SOLID   OF  MINIMUM  RESISTANCE.  6-/ 

where  <:  is  a  constant  depending  upon  the  density  of  the  fluid. 
Let  v'  be  the  velocity  of  the  body  in  the  direction  of  the  axis  x. 
Then 

and  (i)  becomes 

j,ds  =  cv'^^,ds.  (3) 

Let  dz  be  the  surface  of  the  elemental  zone,  described  by  ds. 
Then,  since  dz  =  27ryds,  we  shall  have/</^,  or  the  normal  pres- 
sure upon  this  zone, 

=  27typds  =  271  cv'"^  ^-^  ds,  (4) 

Now  the  force  pdz,  being  distributed  normally  about  the 
zone,  may  be  regarded  as  aggregated  and  exerted  in  the 
direction  of  any  particular  normal,  and  may,  moreover,  be 
resolved  into  two  components,  the  first  in  the  direction  of  x, 
and  the  second  in  the  direction  of  y,  and  the  first  is  the  only 
one  which  resists  the  motion  of  the  body  in  the  direction  of  x. 
Therefore,  rdz  being  the  resistance  of  any  elemental  zone  to 
the  forward  motion  of  the  body,  we  have 

rdz=pd.'t^:=2nc'J^^!^ds,  (5) 

in  which  only  the  numerical"  values  of  the  quantities  have 
been  regarded.  Let  R  be  the  resistance  of  any  zone  included 
between  the  two  planes  ^  =  ^r^,  x  ^  x^,  and  we  shall  have 

R  =  2^c.''  r'l^.  ds  ^  2ncv-  r^^Jx. 

e/so       ds  ^^0     I  -\-  y'^ 

Now  we  shall  not  regard  the  resistance  experienced  by  any 
plane  cylindrical  end,  should  there  be  one,  so  that  R  is  the 


68  CALCULUS  OF  VARIATLONS. 

quantity  which  must  become  a  minimum.  Therefore,  neglect- 
ing, as  usual,  the  constants  c,  tt,  v\  we  are  to  minimize  the 
expression 


y 

Here,  as  Fis  a  function  of  y  andy  only,  and  as 

p_,,3yxi+y')-2/' 
^     (I  +/o=     ' 

we  have,  by  formula  (C),  Art.  56, 

i+y'  ^^     (i+zT  ^^ 

which  is  the  convenient  form  of  the  constant.     Whence  we 
derive  the  equations 

yy"   ^  lyy"  _   .  ^yy" ^, 


I  +/"   (I  +y? 


^-  (7) 


Reducing  the  first  member  of  (7)  to  a  common  denominator 
and  solving  for  y,  we  obtain  $ 

y  =  '^f^.  (8) 

From  this  differential  equation,  although  it  cannot  be  fur- 
ther integrated,  we  may  obtain  the  value  of  x.  For  differ- 
entiate (8),  and  w^e  have 

dy  or  yd. = .  y-4y(i+yvy-(.+/T3y!^.    (^^ 


SOLID   OF  MINIMUM  RESISTANCE.  69 

Whence 

^.  =  .4/'(i+/')73(i+/?^y 


y" 


^,y:^'_^,_,, 


y 

_     idy'  _2dy'  _idy'\ 

~     \y'         y"         y"r  ^^^^ 

which  is  easily  integrated,  giving 

Now  if  we  suppose  equations  (8)  and  (11)  to  be  combined 
so  as  to  eliminate  y' ,  we  shall  obtain  an  equation  between 
X,  y,  c,  and  d,  which  will  be  the  equation  of  the  required 
curve  in  finite  terms,  and  may  be  supposed  to  be  expressed 
under  the  form 

/{x,y,c,d)  =  o=/. 

Then  if  we  suppose  the  values  of  y^  and  y^  to  be  given,  we 
shall  have 

/{x^,  y„  c,  d)  =  o,         /{x„  Jo,  c,  d)  =  0; 

from  which  equations  we  must  determine  c  and  d  in  terms  of 
the  given  quantities  x^,  x^,  y^,  y^.  But  if  y^  anii  y^  be  not  given, 
we  shall  have 


^  =  p.  =y 


,/3/'(i+y')-2y'\_^. 


which  gives  either  y^-=^o   or  jj//  =  o ;  and  a  similar  equation 
for  the  lower  limit. 

But  it  is  easy  to  see  that  the  form  of /cannot  be  practi- 
cally determined,  as  the  elimination  of  y'  just  proposed  can- 


/O  CALCULUS  OF  VARIATLONS. 

not  be  effected  ;  while  it  is  well  known  that  the  theory  which 
we  have  adopted  regarding  the  resistance  of  fluids  is  not 
altogether  trustworthy.  The  problem  will,  however,  afford 
ground  hereafter  for  some  useful  remarks  regarding  the  terms 
of  the  second  order,  and  is  also  of  historic  interest,  having 
occupied  the  attention  of  Newton,  Legendre,  and  others. 

Problem   VII. 

69.  It  is  required  to  detennine  among  all  curves  which  can  be 
drazun  between  two  fixed  poifits^  that  which,  being  revolved  about 
the  axis  of  x,  will  generate  the  surface  of  mi7iimum  area. 

Let  ds  be  an  element  of  the  generating  curve.  Then  the 
surface  which  is  to  become  a  minimum  will  be  27r  /  ^ yds,  or 

e/so 

2nJ  ^ y  \/i  j^y'^^dx,  SO  that,  neglecting  the  constant,  we  must 
minimize  the  expression 


u=£yv,+y^dx=£ydx. 


yy' 


Here    F  is   a   function  of  y  and  /  only,  and  P—  — ===. 
Hence,  by  formula  (C),  Art.  56,  we  have  the  equations 


y      _ 


=  a,  (2) 

Squaring,  clearing  fractions,  and  transposing,  we  have 

y'=^^^.  (3) 


MINIMUM  SURFACE   OF  REVOLUTION,  Jl 

To  render  (3)  integrable,  we  must  solve  thus : 

the  integral  of  which,  using  the  upper  sign,  is 

^^al{y+Vf'^^')-\-b,  .   (5) 

the  equation  of  the  catenary,  as  we  will  next  show. 

Now  it  is  plain  that  if  we  regard  the  axis  of  x  as  fixed,  but 
that  of  y  as  movable,  we  can  render  b  any  quantity  we  please 
by  suitably  choosing  the  position  of  that  movable  axis  ;  that 
is,  by  suitably  determining  the  origin  on  the  fixed  axis  x.  In 
this  case  let  it  be  so  taken  that  b  becomes  —  ala,  and  then 
(5)  will  become 

.  =  ./^±.^ZEZ.  (6) 

Let  e  be  the  Napierian  base,  then  (6)  will  give 


ae^  =y-\-Vy''  —  a\  (7) 

But  from  (3)  we  obtain 


Vy"^  —  a^  ^=  ay\ 
whence  (7)  gives 

X 

a^=y-\-ay'.  (8) 

Now  if  in  (7)  we  make  j/  =  o,  4:  becomes  imaginary. 
Whence  the  curve  does  not  meet  the  axis  of  x,  and  y  is  always 
positive.  But  if  we  make  x  —  o,  y  becomes  a,  and  y'  at  this 
point  becomes  zero,  it  being  zero  at  no  other.     Moreover,  we 

have  y"  =  --,  so  that  the  curve  is  convex  to  the  axis  of  x,  and 
a 

is  without  cusps  or  points  of  inflection.     Therefore  /  changes 

sign  when  ;ir  =  o,  and  also  we  have  certainly  a  minimum  ordi. 


72  CALCULUS  OF   VARIATIONS. 

nate  at  that  point.  Now  as  points  which  have  equal  ordi- 
nates  have  also  y'  numerically  equal,  but  positive  or  negative 
according  as  the  point  lies  at  the  right  or  left  of  the  origin, 
and  as  (6)  shows  that  there  can  be  no  two  equal  values  of  y 
on  the  same  side  of  the  origin,  we  conclude  that  the  curve 
has — at  least  so  far  as  it  extends — for  every  point  at  the  right 
of  the  origin,  a  point  at  the  left,  having  an  equal  ordinate, 
while  the  values  of  x  and  y'  are  numerically  equal,  but  with 
contrary  signs.     Hence  we  may  also  write 

X 

ae    ^  ^^  y  —  ay' .  (9) 

Therefore,  adding  (8)  and  (9),  we  obtain 


^a(e^-\-e     A 


y  =  -ale^+e     «j,  (lo) 

which  is  the  usual  form  of  the  equation  to  the  catenary  when 
the  directrix  is  the  axis  of  x,  and  the  origin  under  the  lowest 
point ;  also,  a  is  the  constant  which  would  in  mechanics  rep- 
resent the  tension  in  the  direction  of  x. 

60.  We  have  already  seen  how  to  dispose  of  the  constant 
3  which  occurs  in  the  general  solution,  and  we  now  proceed 
to  consider  the  remaining  constant  a. 

It  must  be  evident  that  even  when  the  limiting  values  of 
X  and  /  are  given — just  as  when  they  are  not — it  may  happen 
that  no  constants  can  satisfy  the  given  conditions;  that  is, 
that  no  curve  of  the  required  kind  can  be  drawn  between  the 
given  points.  Let  us  first  suppose  that  the  two  points  are 
equally  distant  from  the  axis  of  x,  and  let  x^  =  c  and  y^  =  d. 
Then  (10)  gives 

and  from  this  equation  ti  must  be  found  in  terms  of  c  and  d. 


MINIMUM  SURFACE   OF  REVOLUTION.  73 

But  we  are  chiefly  concerned  in  knowing  when,  if  at  all, 
the  solution  will  become  impossible ;  and  this  point  we  will 
now  consider.  If  we  differentiate  the  second  member  of  (ii) 
under  the  supposition  that  c  is  constant  and  a  variable,  and 
then  equate  the  result  to  zero,  we  shall  obtain,  on  solving,  the 
values  of  a  expressed  in  terms  of  c,  if  any  exist ,  which  will 
render  b  a  minimum.     Performing  this  operation,  we  have 

ij^  +  ^~«)---(^-^"«)  =  o.  (12) 

Developing  each  term  of  (12)  carefully  by  Maclaurin's  Theo- 
rem, we  have 

-etc.  =  0,  (13) 


2a         \4    a 

an  equation  which  evidently  gives  but  one  positive  value  for 

a^  because  its  first  member  is  —  00  when  a  is  zero,  and  unity 

c 
when  a  is  infinite.     But  (12),  when  solved  for  — ,  is  known  to 

a 

c 
give  approximately  -  =  1. 19968  =  d,  which  evidently  renders 

b  a  minimum,  as  it  is  clear  from  (11)  that  we  can  make  it  in- 
finite by  making  a  infinite.  To  determine  this  minimum  value 
of  b  in  terms  of  c,  first  substitute  in  (11)  thus: 


'  =  ;("+?)• 


which,     being    solved,    is    known    to    give    approximately 

-=  1.81017.      Therefore  we  have  -=  1.5088. 
a  c 

Now  as  this  equation  gives  the  least  value  of  b  in  terms  of 
^,  it  is  evident  that  if  the  extreme  points  be  so  given  that  — 
will  be  less  than  1.5088,  there  can  be  no  catenary  drawn  hav- 


74  CALCULUS  OF  VARLATIONS. 

ing  the  axis  of  x  as  its  directrix,  although  of  course  some 

catenary  can  always  be  drawn ;   and  if  —  become  equal  to 

1.5088,  then  a  single  catenary  can  be  drawn  in  which  a  must 

equal or    -— ;   and  if  -   become    p^reater    than 

^         1. 19968  1.81017  c  ^ 

1.5088,  then  two  real  and  positive  values  can  be  found  for  a, 

and  we  may,  by  using  each  in  succession,  draw  two  catenaries 

between  the  two  given  points,  each  having  the  axis  of  x  as  its 

directrix. 

61.  As  it  will  be  found  highly  important,  in  determining 
the  question  of  their  minimum  property,  to  distinguish  be- 
tween the  upper  and  lower  catenary,  we  must  now  also  con- 
sider the  more  general  case  in  which  j/^  and  j/^  are  unequal. 

Suppose,  then,  that  the  given  points  are  unequally  distant 
from  the  axis  of  x,  x  being  so  estimated  that  j/j  shall  be  greater 
than  j^p.  Then  move  the  origin  along  the  axis  of  ;ir  to  a  point 
midway  between  the  ordinates  j/q  and  j/^.  Denote  j^^  by  d, 
y^  by  kj  x^  by  c,  and  x^  by  —  c.  Then  n  being  the  distance 
of  the  new  origin  from  the  former,  and  of  course  positive,  the 
general  equation  of  the  catenary  becomes 


(I) 

Hence  we  have  at  the  limits  the  equations 

/  c  +  n  c  +  n\ 

b=z^le'^+e       ^y  (2) 

(n  —  c  n  —  c\ 

e~^+e       ^j.  (3) 

From  these  equations  we  must  now  find  a  and  n.     Mul- 


MINIMUM  SURFACE   OF  REVOLUTION.  75 

c  _c 

tiply  (2)  by  e^  and  (3)  by  e  ",  and  subtract ;  then  multiply  (2) 

_c  c 

by  e  ^,  and  (3)  by  ^,  and  subtract.     Then  we  shall  have  the 
two  equations 

n    /  2c  _  2c\  c  _  c 

^  ^  /  ^«  —  ^      a  \  =  ^^  _  i^     a^  (4) 


n  /    _  2c  2c\  _  c  c 


Changing  signs  in  (5)  and  multiplying  by  (4),  we  have 

2c  2c\  2 


-ie'^  —  e     "■\  =  ibt^  —  ke    ''\  ik(^  —  be    "  !. 


(5) 


(6) 


Having  thus  eliminated  n,  we  must  now  determine  whether 
(6)  can  be  satisfied  by  any  real  and  positive  value  or  values 
of  a.     Write,  for  convenience, 

2  /  2c  _?£\^  /     c  _9\     /      £  __c\ 

F^-ie'^—e     ^\  —  ib^  —  ke    "A  ik^  —  be    ^\       {?) 

which  becomes  zero  whenever  a  catenary  is  possible.  Differ- 
entiating F  under  the  supposition  that  a  only  is  variable,  we 
obtain 

dF_  C    2c  _2c^ 


l(,l_,-?)-.(J+.-l)  +  !^^|. 


Now  if  F'  can  vanish  for  any  real  and  positive  value  of  a, 
F  has  a  corresponding  minimum  value  or  values.  For  it  has 
its  greatest  when  a  is  zero,  its  value  then  being  infinite.    For 


^6  CALCULUS  OF  VARIATIONS. 

if  we  develop  by  Maclaurin's  Theorem  the  first  member  of 
the  following  equation,  we  shall  have 

.i(,l-.-l)..(,+f-^+$:^_+e..). 

Whence,  when  a  is  infinite,  i^  becomes  4^"  +  (<^  —  ky ;  and  when 
a  is  zero,  F  becomes  infinite. 

Now  to  determine  whether  F'  can  vanish  and  change  its 
sign   as   a   ranges   from   zero   to   positive  infinity,  we  must 

2c  _2c 

recollect  that  e^  —  e  "  is  of  invariable  sign,  and  that  therefore 
the  only  part  of  F'  which  can  change  its  sign  is  the  second 
factor,  and  this,  when  developed  by  Maclaurin's  Theorem 
and  arranged,  becomes 


2  c\  ^  -zz^  [^ ~\-  etc 


2V*  /ji_ i_\ 


(9) 


Now  if  hk  be  greater  than  — ,  we  can  evidently  make  (9),  and 

consequently  F' ,  vanish  and  change  its  sign  once,  and  once 
only,  for  any  real  and  positive  value  of  a ;  and  therefore  F  is 
in  this  case  susceptible  of  a  minimum  value;  and  if  this  mini- 
mum value  be  negative,  F  can  be  made  to  pass  through  zero, 
and  to  change  its  sign  twice.  Hence  in  this  case  equation  (7), 
or  i^=  o,  can  be  satisfied  by  two  real  values  of  a\  and  we  can 
draw  two  catenaries  by  using  these  values  successively.  But 
if  the  minimum  value  of  F  be  zero,  F  can  touch  zero  but  once, 
and  (7)  can  only  be  satisfied  by  one  value  of  a^  and  thus  we 
can  draw  but  one  catenary.  If  the  minimum  value  of  F  be 
positive,  then  F  cannot  become  zero  at  all,  and  (7)  cannot 
be  satisfied  by  any  real  and  positive  value  of  ^,  and  thus  no 
catenary  can  be  drawn. 


SPECIAL   DISCRIMnVATIOX  OF  MAXIMA    AND   MINIMA.      7/ 

Now  if  bk  be  equal  to  or  less  than  — ,  F'  will  be  always 

3 
negative,  and  F  can  consequently  have  no  minimum  value. 
In  this  case  F  cannot  touch  zero  at  all,  and  there  can  be  no 
catenary.  For  we  have  already  seen  that  the  least  value  of 
/^  is  4^'  +  (/^  —  kf,  which  expression  is  evidently  greater  than 
zero. 

62.  The  preceding  article  is  taken  from  Chapter  IV.  of 
the  Adams  Essay,  by  Prof.  Todhunter ;  and  we  shall  now, 
before  closing  this  section,  subjoin  an  investigation  of  the 
terms  of  the  second  order  in  a  particular  class  of  problems, 
in  which  Probs.  VI.  and  VII.  are  included.  This  investiga- 
tion appears  to  be  due  entirely  to  the  same  author.  (See 
Adams  Essay,  Arts.  26,  27.) 


Problem  VIII. 

It  is  required  to  investigate  in  full  the  conditions  which  will 
maximize  or  minimize  the  expression 

where  f  is  any  function  of  y'  only. 

Put  f  for  ^,  and  f"  for  ^.     Then,  to  the  terms  of  the 

ay  dy 

second  order  inclusive,  we  shall  have 

+  \Ly^f'^y  ^y  ■^yf"^y'')dx,  (I) 


78  CALCULUS   OF  VARIATIONS. 

Here  Fis  a  function  of  y  and  y'  only,  and  P=yf',  so  that  we 
have  at  once,  by  formula  (C),  Art.  56,  since  the  terms  of  the 
first  order  must  vanish, 

yf-yy'f  =  ^-  (2) 

This  is  as  far  as  we  can  carry  the  general  solution,  so  long 
as  the  form  of  /  is  entirely  arbitrary,  although  we  may  sup- 
pose the  solution  to  be  of  the  form 

y  =  F{x,  c,  c'). 

63.  Let   us  now   consider   what  transformations  can   be 
effected  in  the  terms  of  the  second  order.     By  parts  we  have 

ff'dy  Sy'dx  =f'6f  -  J  Sy .  ^J'Sy.dx.  (3) 

But 

Whence 

JfSySy'dx  =f'Sf  -ff  6y  Sy'dx  -  /^  6/dx.        (4) 
Therefore 

2ffSy  Sy'dx  =  f'Sf  -/^  Sfdx.  (5) 

Substituting  this  value  in  (i),  and  observing  that  the  terms  of 

the  first  order  vanish,  and  that  -^  =  f"y\  we  have 

ax 

6U  =  l[f:Sy,^  -  f^Sy,")  +L£'(_yf"s/'  _  y"/"Syjdx.    (6) 


SPECIAL  DISCRIMINA  TION  OF  MAXIMA  AMD  MINIMA,       79 

But  if  we  suppose,  as  usual,  the  values  of  y^  and  j,  to  be  as- 
signed, we  have 


^U=-rf\ySy'^-/Sfyx.  (7) 


Now  in  our  applications  of  this  formula  we  shall  usually  be 
able  to  regard  y  as  positive  ;  arid  let  us  also  suppose  that  y" 
does  not  change  its  sign  within  the  range  of  integration ;  that 
is,  that  the  required  curve  is  always,  at  least  for  the  part  that 
we  consider,  convex  or  concave  to  the  axis  of  x.  We  will 
consider  first  the  latter  case.  Here,  since  y"  is  always  nega- 
tive, the  factor  ySy  —  y"Sy^  is  always  positive,  and  therefore, 
\i  f"  be  also  of  invariable  sign,  we  shall  have  a  maximum  or  a 
minimum  acording  as  it  is  negative  or  positive. 

But  we  can  show  that  when  y"  is  of  invariable  sign,  f" 
must  be  also.     For  from  (2)  we  have 

y-j^r  («) 

Whence,  by  differentiation,  we  have 
and  solving  for  y\  we  find 

which  shows  that  when  y"  is  of  invariable  sign,  f"  must  be 
also.  But  since  c  may  be  either  positive  or  negative,  the  sign 
of  f"  cannot  be  determined  so  long  as  the  problem  is  per- 
fectly general,  and  therefore  we  can  only  say  that  when  y"  is 
negative,  we  have  a  maximum  or  minimum  according  as  f" 
is  negative  or  positive. 


80  CALCULUS  OF  VARL4TL0NS. 

Next  suppose  y'^  to  be  always  positive.  Then,  although 
f"  must  be  of  invariable  sign,  we  cannot  say  that  the  factor 
ySy^  -.ySy"  may  not  change  its  sign;  and  therefore  this  case 
will  require  further  investigation,  which  will  be  given  here- 
after, when  we  have  presented  Jacobi's  Theorem. 

64.  We  will  now  apply  the  preceding  formula  to  the  in- 
vestigation of  the  terms  of  the  second  order  in  Prob.  VL, 
although  we  did  not  succeed  in  obtaining  the  equation  of 
the  generating  curve  in  finite  terms. 

From  equation  (9),  Art.  58,  w^e  easily  obtain 

Now  we  will  consider  the  case  in  which  y'  is  positive  and 
of  invariable  sign.  Then,  observing  that  tan^  60°  =  3,  we  see 
that  if  y  be  always  less  than  3,  y'^  will  be  always  negative  ;  if 
y  pass  through  '^, y"  will  change  its  sign;  and  if  y^  be  always 
greater  than  3,  y"  will  be  always  positive.  Hence  we  may  in- 
vestigate the  first  case.  Write  the  fundamental  equation  of 
Art.  58  thus: 


Then  we  find 


and 


,'2 


/"=2y  3    y" 


Hence,  if  we  suppose  the  limiting  values  of  x  and  y  to  be  fixed, 
the  terms  of  the  second  order  will  become 


SHORTEST  CURVE  BETWEEN   TWO    CURVES.  8 1 

which  expression  is  evidently  positive,  thus  giving  us  a  mini- 
mum. 

65.  The  term  minimum  must  here  also  be  understood  in 
its  technical  sense,  and  we  must  by  no  means  say  that  the 
curve  whose  differential  equation  we  have  obtained  is  the 
curve  which  will  generate  the  solid  of  least  resistance.  For 
Legendre  has  shown  that  by  taking  a  zigzag  line  we  can 
make  the  resistance  as  small  as  we  please.  The  fact  is  that  in 
this  case,  as  in  every  other,  we  can,  by  means  of  the  calculus 
of  variations,  compare  the  curve  or  curves  obtained  from  the 
differential  equation  J^f  =  o  with  such  curves  only  as  can  be 
derived  from  their  primitives  by  making  infinitesimal  changes 
in  the  values  of  y  and  y' ,  And  although  we  might  pass  from 
a  continuous  curve  to  a  zigzag  line  by  means  of  infinitesimal 
changes  'v;\y^  we  certainly  could  not  by  such  changes  in  y. 


Section   III. 


CASE  IN  WHICH  THE  LIMITING   VALUES  OF  X  ALSO  ARE 

VARIABLE. 

Problem  IX. 

^^,  Suppose  it  be  required  to  find  the  shortest  curve  which  caji 
be  drawn  so  as  to  connect  two  given  curves,  all  the  curves  lying  in 
the  same  plane. 

Let  ff  and  gg  be  the  given  curves,  and  ab  the  required 
curve,  which  is  of  course  a  right  line.  Then  if  we  assume 
two  other  points,  c  and  d,  indefinitely  near  to  a  and  b,  and  join 
them  by  another  curve  which  is  of  the  same  kind  as,  or 
differs  infinitesimally  in  form  from,  ab,  the  curve  cd  must  ex- 
ceed in  length  the  curve  ab. 


82  CALCULUS  OF  VARLATIONS. 

This  assertion  would  be  equally  true  if  the  points  a  and  c, 
b  and  d  had  not  been  taken  indefinitely  near,  and  if  the  curve 
cd  had  not  differed  infinitesimally  in  form  from  ab.  But  then, 
even  if  cd  were  shown  to  exceed  ab  in  length,  we  could  not 
be  certain  that  some  third  curve  might  not  be  drawn  be- 


tween ff  and  gg  differing  less  in  form  from  ab,  or  having  its 
extremities  a  little  nearer  to  a  and  b,  which  might  be  shorter 
than  either  ab  or  cd. 

Now  since,  whatever  be  its  nature,  the  length  of  the  line 
ab  is  given  by  the  equation 

u  =  n  vT+y^  dx  =  r^  vdx, 

we  see  that  we  are  now  required  to  determine  what  change 
U  undergoes  when  not  only  y\  but  also  the  co-ordinates 
x^,  j/q,  ^1,  Jv  of  the  points  a  and  b,  receive  indefinitely  small  in- 
crements. 

Now  it  is  evident  that  we  may  pass  from  the  curve  ab  to 
the  curve  cd  in  the  following  manner:  First,  without  at  all 
changing  the  form  of  ab,  give  indefinitely  small  increments 
dx^  and  dx^  to  the  original  limits  x^  and  x^,  so  that  the  new 
values  of  these  limits  may  become  the  abscissas  of  the  new 
points  c  and  d,  which  change  would  give  to  the  curve  ab  an 
increment  like  that  which  it  would  receive  by  differentiation. 
Then,  secondly,  vary  y,  supposing  x  throughout  the  new 
limits  to  be  incapable  of  variation.      By  the  change  in  the 


SHORTEST  CURVE  BETWEEN   TWO   CURVES.  83 

limits  we  obtain  the  required  abscissas  of  the  new  extreme 
points,  while  by  varying  y  we  obtain  their  ordinates,  and  also 
any  desired  alteration  in  the  form  of  the  primitive  curve. 

Now  if  we  denote  by  U'  what  f/ becomes  when  we  change 
x^  into  x^  -\-  dx^y  and  x^  into  x^-\-  dx^,  the  form  of  the  primitive 
curve  being  unchanged — that  is,  y  and  y'  being  unvaried — we 
shall  have 

U'=  /  ^^       Vdx 

t/xo-\-dxQ 

Pxj  pxi  +  dx,  /^Xo  +  dxo 

in  which  expression  it  must  be  remembered  that  the  incre- 
ments dx^  and  dx^  are  to  be  taken  either  positively  or  nega- 
tively, according  as  the  abscissae  of  the  new  extreme  points 
are  further  from,  or  nearer  to  —  00  than  those  of  the  original 
points.     But  it  is  evident  that  to  the  second  order 

p^i+dxi  J  f^y\ 

i.  F^-=F.rf.,  +  i(J^|^.r-  (2) 

and  making  the  same  reduction  for  the  lower  limit,  (i)  be- 
comes 

/"=^.-'.-''-'"-+j|©,'"'-{S.*-l 

+.r  "''-'■  (3) 

which  is  true  to  the  second  order. 

Let  us  next  ascertain,  as  far  as  the  terms  of  the  second  order, 
what  change  would  result  to  U^  from  changing  y^  into  y'  -{-  6/. 
Smce  the  integral  in  (3)  equals  U,  the  change  which  will  re- 
sult to  it  will  be  merely  SU,  where  SU  is  to  be  found  to  the 


84  CALCULUS  OF    VARLATLONS. 

second  order,  and  the  terms  of  the  first  order  transformed  as 
hitherto  explained,  so  that  we  need  only  consider  the  terms 
without  the  integral  sign.  The  change  in  the  term  V^dx^,  prol 
duced  by  varying  j/,  dx^  remaining  unaltered,  is  ^V^dx^,  which, 

if  we  put  as  usual  Pfor 4=,  becomes 


the  first  term  only  being  retained,  as  the  others  are  evidently 
of  an  order  higher  than  the  second ;  and  similarly  for  the  cor- 
responding term  at  the   lower  limit.     The  term  —[-—-]  dxJ^ 

2\dx  U 

is  already  of  the  second  order,  and  must  simply  be  retained 
without  regarding  its  variation,  every  term  of  which  would 
be  of  an  order  higher  than  the  second.  Similarly,  we  merely 
retain  the  corresponding  term  at  the  lower  limit. 

Now  collecting  our  results,  and  denoting  by  {pUI  the 
entire  change  which  the  length  of  ab  or  U  has  undergone,  we 
shall  have 

\S  C/]  =  V,  dx,  -  K  dx,  +  />,  Sy,  -  />.  Sy,  +fjj-  ~  Sy  dx 

I        /*^1     I 

+  -Xp<^/V^.  (4) 

67.  Now  it  will  appear,  by  reasoning  in  all  respects  similar 
to  that  which  has  been  hitherto  employed,  that  since  dx^  and 
dx^,  like  Sy^  and  ^y^,  are  capable  of  either  sign,  if  U  is  to  be  a 
maximum  or  minimum,  the  terms  of  the  first  order  in  \pU^^ 
must  vanish,  and  those  of  the  second  must  become  positive 


SHORTEST  CURVE   BETWEEN-    TWO    CURVES.  55 

for  a  minimum  and  negative  for  a  maximum.  Disregarding, 
therefore,  at  present  the  terms  of  the  second  order,  we  have 

lSU^  =  V:dx,-V,dx,+  P,Sy^-P,dy^+J^^    -  ^dy  dx.      (5) 

Now  it  must  be  evident  that  the  curve  sought  can  be  no 
other  than  a  straight  line.  For  suppose  the  points  a  and  b  to 
be  joined  by  any  curve  other  than  a  straight  hne.  Then  even 
if  this  curve  were  shorter  than  any  other  Kne  which  could  be 
drawn  between  the  given  curves,  when  one  or  both  the  ex- 
treme points  a  and  b  were  changed,  yet  we  know  from  our 
previous  investigations  that,  without  changing  these  points, 
it  could  be  still  further  shortened  by  making  it  a  right  line. 
Whence  we  see  that  our  present  problem  must  concern  merely 
the  position  which  this  line  must  have  in  order  to  render  its 
length  a  minimum.  Moreover,  the  term  under  the  integral 
sign  in  (5)  is  just  what  it  would  have  been  had  we  merely  re- 
quired the  curve  of  minimum  length  between  two  fixed  points. 

dP 
Therefore,  since  the  right  line  is  the  general  solution,  — -  will 

dx 

vanish,  and  consequently  the  integral  must  vanish,  thus  leav- 
ing us  with  the  terms  at  the  limits,  which  must  also  be  equated 
to  zero. 

This  mode  of  demonstration  will  probably  be  most  appa- 
rent, but  the  following  is  the  true  analytical  method.  By  rea- 
soning similar  to  that  employed  in  Art.  39  and  the  preceding 
articles,  we  can  show  that  the  term  under  the  integral  sign 
must  vanish,  as  must  also  those  free  from  the  sign  of  integra- 
tion, taken  collectively.  Equating  the  integral  to  zero,  we 
obtain,  as  before,  the  right  Hne  as  the  general  solution,  and 
have  then  to  consider  the  remaining  terms,  which  may  be  rep- 
resented by  the  equation  L—o. 

68.  We  have,  then,  from  (5), 

L  ==  V,dx,  -  V^dx,  -^P,6y,  -  P,dy,  =  o.  (6) 


86  CALCULUS  OF    VARIATIONS. 

Now  if  the  quantities  dx^,  dx^,  6y^^  dy^  were  entirely  indepen- 
dent, we  would  evidently  be  obliged  to  equate  the  coefficient 
of  each  one  severally  to  zero.  Then  we  would  have  four 
equations  at  the  limits  to  be  satisfied,  whereas  the  general 
solution  contains  but  two  arbitrary  constants,  and  this  would 
usually  be  impossible  in  any  problem.  But  in  the  present 
case  we  know,  without  further  investigation,  that  two  of  these 
equations,  V^^o  and  V^  =  o,  cannot  be  satisfied  by  any  real 
value  of  y.  This  is  as  it  should  be.  For  if  the  quantities  ^;i:,, 
dx^,  ^jj/j,  (5>o  were  independent,  the  extremities  of  the  required 
curve  would  be  entirely  unrestricted,  and  we  could  have  no 
maximum  or  minimum,  because  we  could  always  increase 
or  diminish  its  length  at  pleasure.  But  as  in  the  present  case 
the  extremities  of  the  required  curve  are  confined  to  two 
given  curves,  we  can  obtain  a  definite  result,  and  we  now 
proceed  to  show  the  method  of  imposing  this  condition  upon 
the  question. 

69.  Let  y  —f{^)  and  y  =  F{x)  be  the  equations  of  the  two 
given  curves,  and  let  y  be  any  ordinate  of  the  required  curve, 
and  Y  the  ordinate  of  the  derived  curve  corresponding  to  the 
same  value  of  x.  Then  Y^=y^-\-Sy^,  2ind  Y^=iy^-\- dy^]  and 
let  us  consider,  for  example,  the  upper  limit.  It  is  evident 
that  when  we  derive  cd  from  ab,  the  abscissa  of  d  will  become 
x^  -\-dx^,  where  dx^  may  be  positive  or  negative,  and  the  value 
of  its  ordinate  is  evidently  obtained  by  passing  along  the  de- 
rived curve  from  the  point  whose  co-ordinates  are  x^  and  F, 
to  the  point  whose  abscissa  is  x^  -\-  dx^ ;  that  is,  to  the  point  d. 
Denoting  then  the  ordinate  of  d  by  n,  we  have 

n=Y,+  Y,'dx,-^-  Y," dx^  +  etc.  (7) 

Hence,  substituting  in  (7)  the  value  Y^=y^-\-Sy^^  and  omit- 
ting all  terms  of  an  order  higher  than  the  second,  we  have 


SHORTEST  CURVE  BETWEEN   TWO   CURVES.  8/ 

n  =  [y-\-dy  -\-y'dx  +  dy'dx  +  -  y" dx^  ■  (8) 

But  since  n  is  an  ordinate  of  the  given  curve  whose  equation 
is  Y  —  f{x),  we  must  have  n^^  f{x^-\- dx^.  Developing  this 
expression  by  Taylor's  Theorem,  we  have  to  the  second  order 


«=/,+/.'^-^.  +  j/.V-^-A  (9) 


where 


/'=£'    /"=S'    ^-^(^-^ 


Combining  (8)  and  (9),  observing  that_j',  =/„  we  have 

*/.  =  (/'  -/).  dx,  +  i(/"  -/).  dx^  -  6y:dx,  (ID) 

Similarly,  we  have  at  the  lower  limit 

Sy,  =  {F'-y'\dx,-\-^-{F"-y\dx;-6y:dx,.  (10) 

70.  If  now  we  substitute  in  (6)  the  values  of  Sy^  and  dy^ 
just  found,  and  set  aside  all  terms  of  the  second  order,  which 
must  be  added  to  those  of  the  second  order  in  (4),  we  shall, 
after  restoring  the  values  of  V  and  P,  have 


y'f'-y' 


^  =  ^-^TT7+^'+^ 


"1^^. 


Having  thus  eliminated  Sy^  and  Sy^,  it  is  evident  that  the 
remaining  quantities  dx^  and  dx^  are  absolutely  independent, 
and  that  we  must  therefore  equate  their  coefficients  severally 


88  CALCULUS   OF    VARIATIONS. 

to  zero.     Performing  this  operation,  and  reducing,  we  have 
the  equations 

I  ■\-ylf!  =  o,  and     I  +  y.'F:  ^  o  ; 

equations  which  show  that  the  required  right  Hne  ab  must  be 
normal  to  each  of  the  curves  ff  and  gg. 

71.  Although  for  the  sake  of  simplicity  we  have  used 
equation  (6),  it  is  evident  that  the  true  mode  of  reasoning 
would  be  the  following:  First  eliminate  Sy^  and  Sy^  in  (4)  by 
the  use  of  equation  (10),  by  which  elimination  we  shall  add 
some  terms  to  those  of  the  second  order.  Then,  by  the  usual 
reasoning,  those  of  the  first  order  must  vanish.  But  these 
terms  will  then  consist  of  L  as  given  in  (11),  together  with  an 
integral;  and,  by  the  reasoning  already  employed,  these  two 
parts  must  separately  vanish.  Now  by  making  the  integral 
vanish,  we  obtain  the  right  line  as  the  general  solution ;  while 
by  making  L  vanish,  we  obtain  at  once  equation  (11),  from 
which  we  derive  the  same  conclusion  as  before. 

72.  If  we  use  equation  (6),  recollecting  that  it  is  true  to 
the  first  order  only,  we  may  evidently  obtain  the  complete 
terms  of  the  second  order  by  adding  to  those  already  in  (4) 
those  which  result  from  the  elimination  of  Sy,  and  Sy^  in  (6) 
by  the  use  of  equation  (10).  But  these  terms  will  by  either 
method  become,  since  those  of  the  first  order  vanish, 


^Jx,  2(1 +y^)^  -^  ^   ' 

Now  the  integral  in  (12)  is  known  from  Prob.  I.  to  be  posi- 
tive, so  that  we  shall  be  sure  of  a  minimum  if  the  remaining 


SHORTEST   CURVE  BETWEEN    TWO    CURVES.  89 

terms  be  positive,  but  not  otherwise.     But  since  the  solution 

y' 
is  a  right  line,  — -^ is  a  constant,  say  c,  and  these  terms 

1/1  +/' 
become 


[f:dx: 


But  c  is  the  sine  of  the  inclination  of  ab  to  the  axis  of  x,  and 
we  may  therefore  so  assume  this  axis  as  to  render  c  positive, 
and  then  we  shall  be  sure  of  a  minimum  if  f"  be  positive 
while  F^'  is  negative. 

73.  But  it  is  unnecessary  to  pursue  this  investigation  fur- 
ther. For  it  must  now  appear  that  the  problem  under  con- 
sideration is  one  rather  of  the  differential  calculus  than  of  the 
calculus  of  variations.  For  since  we  know  from  Prob.  I.  that 
the  right  line  is  the  plain  curve  of  minimum  length  between 
two  points,  whether  they  be  situated  upon  given  curves  or  not, 
we  might  have  been  certain  beforehand  that  the  solution  could 
be  no  other  curve  than  the  right  line,  and  that  our  problem 
could  concern  nothing  but  its  position.  Moreover,  its  posi- 
tion being  determined,  we  need  only  compare  the  line  with 
other  right  lines  drawn  to  points  on  ff  and  gg  consecutive  to 
a  and  b.  For  if  we  vary  ab  so  as  to  obtain  a  derived  curve, 
cd,  which  is  not  exactly  a  right  line,  then,  even  if  we  show  that 
ab  is  shorter  than  cd,  we  could  shorten  cd  by  making  it  a 
right  line,  its  extremities  remaining  unchanged,  and  could  not 
without  a  new  comparison  be  certain  that  the  new  line  cd 
might  not  be  shorter  than  ab. 

The  problem  might  then  have  been  enunciated  thus :  To 
find  the  position  of  the  right  time  of  miiiimtun  IcngtJi  which  can  be 
drawn  between  two  given  plane  curves. 

74.  Although  problems  of  this  sort  might  be  altogether 
omitted  here,  there  appears — at  least  so  far  as  the  terms  of  the 
first  order  are  concerned — to  be  some  advantage  in  solving 


90  CALCULUS  OF  VARIATIONS. 

them  by  the  calculus  of  variations  instead  of  by  the  ordinary 
methods  of  maxima  and  minima.  At  all  events,  they  are  gen- 
erally discussed  by  writers  on  this  subject,  and  it  is  deemed 
necessary  to  render  the  reader  familiar  with  the  methods 
which  they  employ.  We  shall  therefore  subjoin  a  few  more 
problems  of  the  same  kind,  considering  the  terms  of  the  first 
order  only,  since  a  discussion  of  those  of  the  second  would  in 
general  be  unsatisfactory. 

Problem  X. 

76.  It  is  required  to  determine  both  the  nature  and  position  of 
the  curve  which  will  minimize  the  time  of  descent  of  a  particle 
from  one  given  curve  to  another,  the  particle  starting  from  a  fixed 
horizontal  line,  and  being  acted  upon  by  gravity  solely,  all  the  curves 
lying  in  the  same  plane. 

Assume  the  fixed  horizontal  line  as  the  axis  of  x,  and  let 
x^  and  Jo  be  the  co-ordinates  of  the  point  in  which  the  required 
curve  cuts  the  upper  of  the  given  curves,  while  x^  and  y^  are 
the  co-ordinates  of  the  point  in  which  it  cuts  the  lower.  Then, 
reasoning  as  we  did  in  Prob.  II.,  we  see  that  we  have  to  min- 
imize the  expression 

Now  it  is  clear  that  here,  as  in  the  preceding  problem,  the 
limits  of  integration  will  be  also  subject  to  variation.  For 
suppose  that  after  the  required  curve  and  its  points  of  inter- 
section with  the  given  curves  have  been  found,  we  assume 
points  on  the  given  curves  consecutive  to  those  just  found, 
and  then  connect  these  new  points  by  another  curve.  Then 
the  abscissse  of  these  new  points  will  be  x^  +  dx^  and  x^  -\-  dx^, 
dx,  and  dx,  having  either  sign.  It  also  appears,  as  before, 
that  the   total  change  which    U  will  undergo,  both  from  a 


BRACHISTOCHRONE   BETWEEN   TWO    CURVES.  QI 

change  in  the  form  of  the  curve  and  an  alteration  in  the  posi- 
tion of  its  extremities,  can  be  found  by  first  changing  the 
Hmits  of  the  integral  in  such  a  manner  that  the  new  limits 
may  be  the  abscissae  of  the  new  points,  while  the  form  of  the 
curve  remains  unaltered,  and  then  changing  by  the  ordinary 
methods  of  variations  the  form  of  the  curve  taken  throughout 
the  new  limits.  By  the  change  of  limits  only,  U  becomes  U' , 
where  U'  is  given  by  equation  (i)  of  the  preceding  problem, 
because  that  equation  will  hold  irrespectively  of  the  form  of 
V.  Then  if  in  U'  we  change  y  into  y  -\-  ^y,  and  y  into  y'-\-  Sy\ 
and  subtract  W,  we  shall  have  the  exact  value  of  [^U']y  to 
which,  however,  we  can  onl}^  approximate.  This  approxi- 
mation, so  far  as  U'  is  concerned,  is  effected  as  in  equations 
(2)  and  (3)  of  the  last  problem,  which  also  hold  irrespectively 
of  the  form  of  V.  If  now  we  take  the  variation  of  W^  in  the 
usual  way,  we  shall  have  first  the  terms  <^V^dx^  —  SV^dx^, 
which  are  evidently  of  the  second  order  and  must  be  rejected 
unless  we  are  developing  [<^U']  to  the  second  order,  when 
they  must  be  added  to  those  involving  dx^'^  and  dx^"".     Next  we 

obtain  d6^  or    /    'dVdx,  where  c^^  is  to  be  developed  to  the 

first  or  second  order  as  required,  and  the  terms  of  the  first 
order  transformed  as  in  the  case  of  fixed  limits.  Hence  to 
the  first  order  we  shall  have 

[d  ^]  =  V,  dx,  -  V,  dx,  +  P'd  Vdx, 

which  equation  would  evidently  hold  irrespectively  of  the 
form  of  V. 

But  as  in  the  present  case  F  contains  jk  andy  only,  if  we 

put  as  usual  N  for  -—  and  P  ior  — -,  and  then  develop  dF  to 

ay  dy  ^ 

the  first  order,  and  transform  as  usual,  we  shall  have 

[SU^  :=  V,  dx,~  V,  dx.+Pfy-Pfy^^^'^'  I  ^V  -  g  [  Sydx.  (I) 


92  CALCULUS  OF  VARIATIONS, 

76.  Now  it  will  be  remembered  that  the  relation  expressed 
in  either  equation  (lo)  of  the  preceding  problem,  not  having 
been  established  upon  any  particular  supposition,  is  true  what- 
ever be  the  equations  of  the  limiting  curves.  In  this  case, 
therefore,  if  we  assume  j  —f{x)  and  y  —  F{x)  to  be  the  equa- 
tions of  the  two  given  curves,  we  can  eliminate  dy^  and  dy^^ 
the  terms  of  the  second  order  which  result  from  the  elimina- 
tion being  added  to  those  already  existing,  or  else  being  re- 
jected if  terms  of  the  second  order  are  not  to  be  considered. 
When  these  terms  are  to  be  neglected,  equations  (lo)  are 
better  written 

^y.  =  (/-/).  ^^.         ^lo  =  {F'-y%dx,.  (2) 

Performing  this  elimination,  we  have 

[cJt/]  =.  (^v+P/'-P/\d.r-  {V+PF'-P/ldx„ 

Now  having  equated  the  terms  of  the  first  order  to  zero,  it 
will  appear  that,  as  the  integral  cannot  be  made  to  depend 
upon  terms  which  relate  solely  to  its  limits  without  in  some 
manner  restricting  the  generality  of  the  function  dy,  we  can 
only  satisfy  the  equation  ldU']=o  by  equating  the  integral 
and  the  terms  at  the  limits  separately  to  zero. 

It  will  be  seen  from  (3)  that  [^  U]  and  (5^^  differ  only  in  the 
terms  at  the  limits,  the  integrals  being  identical,  and  this  would 
be  the  case  if  F  were  any  function  whatever  of  ^,  y,  y\y",  etc. 
Hence  if  we  make  the  integral  in  (3)  vanish,  it  must  lead  to 
the  same  general  solution  as  though  we  had  been  discussing 
the  problem  of  the  brachistochrone  between  fixed  points,  and 
therefore  the  general  solution  must  be  a  cycloid. 

It  is  clear,  also,  that  if  dx^  and  dji\  be  entirely  independent,  as 
they  are  in  this  case,  we  can  only  make  the  terms  at  the  limits 


BRACHISTOCHRONE   BETWEEN   TWO    CURVES.  93 

certainly  vanish  by  equating  severally  to  zero  the  coefficients 
of  these  quantities.  Performing  this  operation,  and  substitut- 
ing the  values  of  V  and  P,  we  obtain  for  the  upper  limit 


x/:^zi_+i^i±z:Uo, 


whence  by  reduction  we  have 

and  in  like  manner^  at  the  lower  limit,  we  find 

equations  which  show  that  the  cycloid  must  cut  each  of  the 
two  given  curves  at  right  angles. 

77.  We  see,  then,  from  the  preceding  examples,  that  if 
we  wish  to  determine  the  conditions  which  will  maximize  or 
minimize  any  single  definite  integral  in  which  the  limits 
also  are  to  be  subject  to  an  indefinitely  small  change,  we 
have  merely  to  put  the  integral,  if  possible,  under  the  form 

U^^  I   ^Vdx,  F  being  some  function  of  x,  y,  /,  y\  etc.,  and 

then,  if  the  general  solution  be  known  in  the  case  in  which 
the  limits  are  fixed,  we  need  only  consider  the  terms  at  the 
limits,  as  the  general  solution  will  in  every  case  be  the  same, 
whether  the  limits  be  fixed  or  variable.  Moreover,  if  we  wish 
to  consider  the  terms  of  the  first  order  only,  the  terms  at  the 
limits  in  \SU^  =0  Avill  be  identical  with  those  which  occur  in 
SU  —o,  with  the  addition  of  the  terms  V^dx^—  V^dx^.  Then  if 
no  restriction  be  imposed  upon  the  quantities  dx^,  dx^,  Sy^,  Sy^^ 
the  coefficients  of  these  quantities  must  be  equated  severall}^ 
to  zero.  This  would  give  us,  in  addition  to  the  usual  2n  con- 
ditions, Fj^r  o  and  Fo  ==  o,  equations  which,  as  we  have  already 
seen,  could  not  in  general   be  satisfied,  as  we  would   have 


94  CALCULUS  OF  VARLATLONS. 

2n-\-  2  equations  and  only  2n  arbitrary  constants.  But  when 
the  extremities  of  the  required  curve  are  restricted  to  two 
given  curves,  we  can  eliminate  Sy^  and  ^y^  as  already  shown, 
and  thus  the  number  of  ancillary  equations  is  reduced  once 
more  to  2n. 

Problem  XL 

78,  It  is  required  to  determine  the  conditions  which  must  hold 
at  the  limits^  when  in  Prob.  III.  we  also  demand  that  the  required 
curve  shall  have  its  extremities  upon  two  given  curves. 

Assume,  as  before,  the  differential  equations  of  the  curves 
to  be  dy  =f'dx,dy  :=  F'dx.  Then,  following  the  last  ar- 
ticle, we  neglect  all  terms  except  those  at  the  limits,  since 
the    general    solution    is   known    to    be    a    cycloid.      Here 

(r\       .i;^2\2 

V=-  —    ~^  -— ,  and  the  terms  at  the  limits,  as  will  be  seen 

y 

from  Art.  30,  will,  after  adding  V^  dx^  —  V^  dx^,  become 
_  /4/(i  +/')  ,  A  ('+/T\  s^ 

But  from  equation  (11),  Art.  30,  we  have 

4/(1+/")  I    ^(i+ZT^     , 

/'  '^  dx        f 

Moreover,  we  shall  assume  that  the  cycloid  has  cusps  at  the 
points  whose  co-ordinates  are  suffixed,  in  consequence  of  which 


*    PROB.    VII.    WITH  VARIABLE  LIMITS.  95 

y"  will  become  infinite,  and  the  terms  in  (i)  which  are  divided 
\yy  y"  will  vanish.     Hence  (i)  becomes 

L=-  c{py,  -  Sy,)  =  o.  (2) 

But  ^y^  =  {f'  —  y')xdx^,  ^y^  —  {F'  —  y')^dx^,  and  substituting 
these  values  in  (2),  and  equating  severally  to  zero  the  coeffi- 
cients of  dx^  and  dx^^  we  have 

f:-y:  =  o,     F:-y:=o. 

But  yl  and  y^  are  equal,  because  the  tangents  to  the  cycloids 
at  its  cusps  are  parallel,  and  therefore  the  quantities  jr/,  y^, 
//,  F^  are  equal.  Hence  we  conclude  that  the  chord  joining 
the  two  cusps  of  the  cycloid  must  be  normal  to  each  of  the 
given  curves. 

Problem  XII. 

79.  //  is  required  that  the  generating  ctirve  in  Prob.  VII .  shall 
have  its  extremities  upon  two  given  curves. 

Let  the  equations  of  the  given  curves  be  as  in  the  preced- 
ing problem.  Then  V  =  y  Vi  +y^  and  the  terms  at  the  limits 
become 


L  =  {yVi  +y\  dx-  (y  Vi  +y\  dx, 

_|_  (^Z_\  sy^  _  /    /'^'     \  dy^  =  a  (I) 

Eliminating  Sy^  and  fy^  as  before,  we  have,  after  equating  to 
zero  the  coefficients  of  dx^  and  dx^. 


9^  CALCULUS  OF  VARLATIONS, 

Whence  reducing,  we  have 

which  show  that  the  catenary  must  cut  its  limiting  curves  at 
right  angles.     • 


Section  IV. 


CASE  IN  WHICH  SOME  OF  THE  LIMITING   VALUES  OF  X,  Y,  V, 
ETC.,  ENTER  THE  GENERAL  FORM  OF  V. 

Problem  XIII. 

80,  //  is  required  to  determine  the  nature  and  position  of  the 
curve  down  which  a  particle  will  descend  in  a  minimum  time  from 
one  given  curve  to  another,  all  the  curves  being  in  the  same  vertical 
plane,  and  the  motion  of  the  particle  beginni?ig  at  the  point  of  its 
departure  from  the  upper  curve. 

Assume  the  axis  of  y  vertically  downward,  and  let  x^,  y^, 
x^,  j/j  be  the  respective  co-ordinates  of  the  initial  and  terminal 
points  of  the  motion,  and  let  the  differential  equations  of  the 
respective  curves  be  ^  =  F'dx,  and  dy  =/V;r.  Now  in  this 
case  the  velocity  of  the  particle  at  any  point  whose  ordinate 
is  y  will  be  \^2g{^y  —  y^,  because  the  motion  begins  at  the 
point  whose  ordinate  is  j,.  Therefore  in  this  problem  we 
must  minimize  the  expression 


U^r^±£^dx=rVd:c.  (I) 

t/xo       y  y  —  y  t/^o 

Although  it  at  once  appears  that  the  limits  x^  and  x^  will 
also  be  subject  to  change  in  this  problem,  we  see  that  one  of 
these  limiting  co-ordinates,  jf,,,  enters  likewise  as  a  component 
part  of  F  throughout  the  integral,  and  this  fact  will  require 


BRACHISTOCHRONE    CONTINUED.  97 

some  modification  of  our  previous  method  of  solution,  because, 
since  y^  is  a  component  of  F,  any  change  in  the  value  of /„  will 
produce  a  change  in  that  of  V  throughout  the  entire  range  of 
integration.  Moreover,  the  co-ordinates  at  the  lower  limit 
must  always  satisfy  the  equation  y^  =  F(x^y  so  that  when 
we  change  x\  into  x^  -\-  dx^,  we  necessarily  change  y^  into 
F{x^  -|-  dx^.  It  happens  that  V  is  not  affected  by  any  change 
in  the  values  of  the  other  limiting  co-ordinates,  as  they  do  not 
occur  in  V\  but  if  they  did,  the  method  of  treatment  .would 
be  similar  to  that  which  we  are  about  to  explain  for  y^. 

Now  let  V  be  what  V  becomes  when  we  change  y^  into 
J^o  +  '^o)  ai^d  we  shall  have,  from  the  change  of  limits  only, 

U'=  /  V'dx,  (2) 

If  we  next  change  j/  into  j/  +  ^J>  ^^d  y'  intoy-^  Sy\  and  sub- 
tract U  or  J     Vdx,  the  result  will  be  the  exact  variation  of  U, 

to  which  we  will  now  approximate  as  far  as  terms  of  the  first 
order  only.     As  before,  to  the  first  order,  (2)  becomes 


U 


'^  V/dx-  V:dx,+  r  V'dx,  (3) 


Now  when  we  change  x^  into  x^  +  dx^,  we  to  the  first  order 
change  y^  into  y^  -(-  F^dx^,  and  therefore  V  is  what  V  becomes 
when  we  change  y^  mto  y^  +  F^dx^,  y  and  y'  in  V  being  re- 
garded as  constant,  since  they  in  no  manner  depend  upon  y^ ; 

and  this  change  in  V  will  evidently  be  -—  F^dx^,  where  F^dx, 

dy, 
has  simply  been  put  for  dy^.     Hence  to  the  first  order, 

^'=^+^^.''^-.-  (4) 

Substituting  this  value  in  (3),  rejecting  again  all  terms  of  the 


98  CALCULUS  OF    VARLATIONS. 

second  order,  and  observing  that  F^'  and  dx^  may  be  regarded 
as  constants,  we  have 

U'^  K  dx,-  K  dx,+  F:dxX'^-  dx  +y"'  Vdx.  (5) 


If  now  we  vary  y  and  y' ,  we  shall  obtain  the  variation  of 

/     Vdx  or  U  in  the  usual  manner  for  fixed  limits,  while  the 

variations  of  all  the  other  terms  must  be  neglected,  being  of 

an  order  higher  than  the  first.     Hence  putting  N  for    — -, 

Pfor  -— ,  we  shall  have,  after  the  usual  transformation  of  SU, 

ay 

[<J£/]  =  F.  dx,  -  K  dx,  +  Pfiy,  -  Pfiy,  +  FJdxJJ^'^  dx 

But  —  = =  —  TV,  as  will  readily  appear  from  the  form 

dy,  dy 

of  V  given  in  (i),  so  that  we  have 

\SU-\  =  V,dx-  V,dx,-\-Pfiy-  P,dy-  FJdxJJ^dx 

-{-fJ'M  dy  dx  =  o,  (;) 

dP 

where  M  —  N .     Now  whether  we  can  integrate  the  ex- 

dx 

pression  JJ'Ndx  or  not,  we  know  that  it  is  merely  a  function 

of  the  limiting  co-ordinates  and  their  differential  coefficients, 
the  form  of  the  integral  being  dependent  upon  the  nature  of 
the  general  solution  obtained  by  making  the  second  member 


BRACHISTOCHRONE    CONTINUED.  99 

of  (7)  zero,  and  it  is  not,  therefore,  in  our  power.     Hence,  by 

the  same  reasoning  as  before,  we  must  have  J^    M  Sy  dx  =  o 

and  M—o. 

As  we  have  merely  assumed  that  the  axis  of  y  shall  be 
vertical,  we  may  take  that  of  x  so  as  to  make  y^  zero,  in  which 
case  the  equation  M  —o  will  become  identical  with  the  same 
equation  in  Prob.  II.,  Case  2,  and  the  general  solution  will 
therefore  be  a  cycloid  —  which  solution  will  evidently  also 
hold  however  we  assume  the  axis  of  x^  since  by  changing 
that,  so  long  as  that  of  y  is  vertical,  we  change  neither  the 
form  of  the  curve  nor  the  values  of  any  of  the  differential  co- 
efficients of  y.  The  general  solution  then  being  a  cycloid  with 
a  cusp  on  the  upper  curve,  we  must  next,  if  possible,  satisfy 
the  equation 

\d  U^  =  F,  dx,  -  V,  dx,  +  Pfy^  -  Pfy,  -  F:dx,  r^Ndx  =  o.  (8) 

dP 
Now  in  this  case  the  equation  M=  o  gives  N=  — -,  where  the 

differential  is  total.     Hence 

-  F/dx,  £l'Ndx  =  -  F:dx,  (P,  -  p.).  (9) 

Now  substitute  this  value  in  (8),  and  next  eliminate  Sy^  and 
Sy^  by  equations  (2),  Art.  ^6.  Then  equating  to  zero  severally 
the  coefficients  of  dx^  and  dx^^  we  have 

v,-^pj:-p,y:=o,  (10) 

v,^p,F:-p„y:=o.  (II) 

Since  the  general  solution  is  a  cycloid,  we  have,  from  the 
equation  _>^( I  +y^)  =  ^  of  Art.  25,  by  putting  j—jTo  for  j. 


^U  —  7o)  (i  +/')  ^^a  ^    V2r. 


lOO  CALCULUS  OF    VARLATIONS. 

Substituting  this  value  in  (lo)  and  (ii)  after  having  restored 
the  values  of  V  and  P,  they  become  after  reduction 

equations  which  show  that  the  cycloid  cuts  the  lower  curve 
at  right  angles,  while,  since  //  =  /^/,  the  tangents  of  the  two 
given  curves  at  the  initial  and  terminal  points  of  the  motion 
must  be  parallel. 

81.  We  have  seen  that  when  a  particle  starts  from  a  state 
of  rest,  the  cycloid  must  have  a  cusp  at  that  point.  But  if  it 
is  to  start  with  a  given  initial  velocity  in  the  direction  of  the 
tangent,  which  velocity  could  always  have  been  produced  by 
faUing  from  some  height  //,  Fin  Prob.  II.,  Case  2,  would  be- 
come 

\^y  +  // 
If,  as  usual,  we  obtain  the  differential  equation 

dP 
ax 

we  can  evidently,  while  keepings  vertical,  remove  the  axis  of 
X  to  the  height  h  above  the  initial  point,  without  affecting  the 
form  of  the  curve  given  by  the  equation  M  —  o.  But  making 
this  change,  y  -{-  h  will  become  jk,  and  J/ will  become  identical 
with  M  in  Prob.  II.,  Case  2,  thus  giving  us  a  cycloid  with  its 
cusps  upon  the  new  axis  of  x.  That  is,  when  the  particle 
starts  with  a  given  tangential  velocity,  the  curve  of  quickest 
descent,  or  the  brachistochrone,  will  still  be  a  cycloid,  but 
having  its  cusps  upon  the  horizontal,  from  which  the  particle 
must  have  fallen  in  order  to  acquire  the  given  initial  velocity 
upon  reaching  the  starting-point. 

In  like  manner,  in  the  last  problem,  if  we  require  the  par- 


BRACHISTOCHRONE    CONTINUED.  lOI 

tide  to  start  from  the  upper  curve  with  a  fixed  tangential 
velocity,  due  to  some  height  h,  V  will  merely  become 


and  no  change  will  be  effected  in  the  results  of  the  last  article, 
except  that  the  cycloid  will  no  longer  have  a  cusp  upon  the 
upper  curve,  but  its  cusps  will  then  be  upon  the  horizontal 
whose  distance  above  the  upper  intersection  is  /i. 

82.  As  examples  of  the  kind  discussed  in  the  preceding 
problem  are  not  numerous,  we  shall,  as  a  means  of  more  fully 
developing  the  method  therein  explained,  now  examine  the 
terms  of  the  second  order. 

For  greater  simplicity,  change  .the  independent  variable, 
assuming  the  axis  of  x  vertically  downward ;  and  for  greater 
generality,  suppose  the  particle  to  start  from  the  upper  curve 
with  an  initial  tangential  velocity  due  to  the  height  /i.  Also 
let  the  equations  of  the  curves  he  f  =  ^ W  =  ^  for  the  upper, 
and  f  —  f{x)  —  fior  the  lower,  while  x^,  y^,  ^^,  f,,  are  the  co- 
ordinates of  the  initial  and  terminal  points  of  the  motion. 
Now  we  shall  have 

Let  F'  be  at  once  what  V  becomes  when  /  is  changed  into 
y  +  ^/y  and  X,  into  x^  -\-  dx,.     Then  we  have 

'  [^^]  =X+<..  ^'^-  -X  ^^-'  (0 

which  is  exact ;  and  we  will  now  approximate  to  the  second 
order.    We  have 


102  CALCULUS  OF    VARLATLONS. 


J..,  dV      ,        dV   r 


For  brevity,  let  A  denote  all  the  terms  of  the  first  order  ex- 
cept F,  B  those  of  the  second,  and  C  their  sum.  Then  (i) 
becomes 

px-i^  +  dxx  f*Xy  pxi  +  dxi 

['^^]=X  +  ...    Vd.-l^    Vd.^l^^^^  Cdx.  (3) 

But,  as  formerly,  the  first  integral  in  (3)  gives 

F.^..-F,^..+  l{f)_^./-l(g')^^.;  +  XV^..     (4) 

Moreover,  neglecting  terms  of  an  order  higher  than  the 
second,  the  last  integral  in  (3)  becomes 

nx^  +  dx^  nxi 

l+...^d.-^lBd..  (5) 

Also  to  the  second  order 

nxi  +  dx^  pxi 

/    ,^      Adx  =  A,dx^-\-A,dx,-\-        Adx.  (6) 

JxQ-\-dxQ  ^        ^     '         °        °     '  t/Xo  ^    ' 

Hence,  finally,  we  have 

+  A,dx,-A,dx,-\-£ydx.  (7) 

Restoring  the  value   of  A,  transforming  by  integration,  as 


BRACHISTOCHRONE    CONTINUED.  IO3 

usual,  the  term   /   '-^,  Sy'dx,  and  then  eUminating  dy^  and  Sy^ 

*^^o   ay 

by  equations  (10),  Art  69,  we  have 

'+(f)y'-^).''--(f).('---'V^.+^.rj^'" 

+£'BJ..  (8) 

Making  the  terms  of  the  first  order  vanish,  we  shall,  as  before, 
obtain  the  cycloid  as  the  general  solution,  and  it  will  be  sub- 
ject to  the  conditions  already  explained.  Then  [^f/]  will 
consist  of  the  terms  of  the  second  order  only,  which  must 
become  positive  if  the  solution  give  a  true  minimum.  As  the 
terms  in  Sy  cancel,  we  shall  have 

V     +£'^^^-  (9) 


104  CALCULUS  OF  VARIATIONS. 

Now  we  cannot  render  it  evident  that  this  value  of  \pU'\  is 
necessarily  positive,  nor  will  any  of  our  subsequent  investiga- 
tions afford  us  the  required  assistance,  there  being  no  known 
method  apphcable.  Therefore,  although  the  great  Legendre 
erroneously  supposed  that  we  were  sure  of  a  minimum,  we 
cannot  in  fact  be  certain  of  its  existence  in  every  case.  (See 
Todhunter's  History  of  Variations,  Arts.  202,  300.) 

83.  When  V  contains  several  of  the  quantities  x^,  jKo,  y^, 
x^,  y^,  j/,  etc.,  the  expression  for  [_SU']  becomes  somewhat 
complicated.  But  as  we  know  that  to  the  first  order  the 
change  which  any  function  undergoes  from  an  indefinitely 
small  alteration  in  any  of  its  components  may  be  found  by 
considering  each  change  separately  and  then  taking  their 
sum,  we  may,  as  Prof.  Jellett  has  suggested,  use  this  method 
with  advantage  here,  as  we  shall  not  require  the  terms  of  the 
second  order. 

Suppose,  then,  that  we  have  to  maximize  or  minimize  the 

expression  U  —  j     Vdx,  where  F  is  a  function  of  x,  y,  y',  and 

also  some  of  the  limiting  values  of  these  quantities,  x^  and  x^ 
being  subject  to  change  into  x^  -\-  dx^  and  x^  +  d^\-  Fi'on^  the 
change  in  x^  alone,  supposing  the  other  quantities  could  re- 
main unaltered,  U  will  be  increased  by  V^  dx^  —  V^  dx^.    From 

varying  y,  /,  etc.,  V  would  be  increased  by  -1-  ^y  ~\~  ~ri  ^y'  H~ 

etc.,  or  6V,  and  U  would  therefore  be  increased  by  J^    SVdx. 

Lastly,  by  the  alteration  in  the  limiting  quantities  which  enter 
it,  F  would,  throughout  the  entire  range  of  integration,  be  in- 
creased by  ^  dx,  +  ^(3j/,  -f  4^  (5>/  +  etc.,  and  the  same  for 
•^  dx,  dy^  dy^ 

the  lower  limit.  Calling  this  change  S'  V,  U  is  increased  by 
J     d'  Vdx.     Adding  these  results,  we  have 


MIXED  INTEGRALS.  105 

\SU^  =  V,dx,  -  VJx,  +  £y'Vdx  +£yVdx  =  o.     (I) 

Now  the  last  integral  in  (i),  being  transformed  as  usual, 
will  give  us,  besides  certain  additional  terms  at  the  limits,  a 
differential  equation  M  =  o,  and  this  equation  will  be  the 
same  in  form  as  though  V  had  not  contained  any  of  the  limit- 
ing components.  Hence  the  general  solution  will  be  the 
same  as  though  Khad  not  contained  these  quantities,  and  the 
limits  also  had  been  fixed.  Then,  by  using  this  general  solu- 
tion, we  must  if  possible,  by  definite  integration,  express  the 
remaining  integral  in  terms  of  suffixed  quantities,  our  power 
to  complete  the  solution  being  dependent  upon  our  ability  to 
remove  this  integral  sign.  After  this  has  been  done,  we  dis- 
cuss the  resulting  limiting  equations  as  we  would  in  any  other 
case. 


Section  V. 

CASE  IN  WHICH  U  IS  A  MIXED   EXPRESSION;    THAT  IS,   CON- 
TAINS AN  INTEGRAI,  TOGETHER  WITH  TERMS  FREE 
FROM  THE  INTEGRAL  SIGN. 

Problem  XIV. 

S^.  //  is  required  to  maximize  or  minimize  the  expression 

U  =  /"""y  ^dx=    r^  Vdx, 

Here  Fis  a  function  of  y,  /,  y\  whence,  by  formula  (A), 
Art.  56, 

v=c  +  py+Qy-y^,  (I) 


and 


I06  CALCULUS  OF    VARIATIONS. 

Hence  (i)  gives 

and,  by  integration, 

y^^cx^d.  (3) 

Now  the  terms  at  the  upper  Hmit  are 

(^-fh  +  a^^'- 

and  similar  terms  at  the  lower  limit,  so  that  unless  some  re- 
striction be  imposed  upon  the  independence  of  Sy^^  6y^,  Sy^\ 
and  SyJ,  there  will  be  four  limiting  equations  to  satisfy,  while 
the  general  solution  contains  but  two  arbitrary  constants,  and 
this  will  in  general  be  impossible. 

But  the  above  example,  containing  the  first  power  only  of 
y",  the  highest  differential  coefficient  in  F,  is,  as  will  be  re- 
membered, a  case  of  Exception  2,  Art.  51.  .  It  will  also  be 
remembered  that  it  was  shown  by  Euler's  method,  equation 
(8),  Art.  52,  that  all  such  integrals  can  be  reduced  to  a  lower 

order,  the  expression  taking  the  form   W^  —  W^-\-  J     V'dx^ 

a  class  of  problems  not  yet  considered.  In  the  present  case, 
recollecting  that  y"dx  =  dy\  y'dx  =  dy,  we  easily  obtain,  by 
parts. 


jT'^y  ^~  dx  =  y^^ly;  -  y.^ly:  -  f\y^  -  ^y'ly'dx 

=  W,-W,+  r'V'dx,  (4) 

Now  if  we  vary  y^,  y/,  we  shall  increase  W^  by 

^  tfr.  +  ^ *r/,      or  {nr-'  iy'Sy\  +(^ sy) , 


MIXED  INTEGRALS.  IO7 

and  we  can  change  W^  in  no  other  way.  A  similar  equation 
of  course  holds  for  W^.  But  these  terms,  relating  to  the  limits 
only,  can  have  nothing  to  do  with  the  form  of  any  general 
solution,  which  must,  therefore,  depend  solely  upon  the  form 
of  V\ 

Now  V  is  a  function  oi  y  andy  only,  and 

P=  —  ny'^  -^ly'  —  ny^  -  ^ 
Hence  by  formula  (C),  Art.  56,  we  have,  as  before, 

^yn  -  \y'  =.  C,  y'*^  z=z  CX-\-  d. 

Now  the  terms  at  the  limits  resulting  from  the  variation  of 
V^  are  P^^y^  —  P^^y^,  which  must  be  added  to  those  obtained 
by  varying  W^  and  W^.  Performing  this  operation,  these  terms 
become 

-  {nyn-^\Sy,^{ny—\dy,+(^)  dy,'  -(^)  (5>;. 

But  these  terms  are  the  same  as  those  which  we  obtained  by 
discussing  the  problem  as  originally  given ;  and  as  the  general 
solution  is  also  the  same,  the  difficulty  which  formerly  oc- 
curred is  not  removed. 

85.  We  may,  however,  from  this  example  see  how  to  pro- 
ceed in  more  important  cases  of  mixed  integrals  which  will 
hereafter  occur.  Thus,  suppose  we  have  to  maximize  or  min- 
imize the  expression 


U=W,-W,+  r^Vdx, 


vxo 


where  W,  and  M^,  are  any  functions  of  x„  y,,  y/,  etc.,  and  x,, 
Jo>  fJ'-'-  y"'^  and  V  is  any  function  oi  x,  y,  /  .  .  .  .  jj/H 
while  the  limiting  values  of  x  are  also  variable.     As  before, 


I08  CALCULUS  OF   VARIATIONS. 

if  we  change  x^  into  x^  -\-  dx^  and  vary  y^,  j//,  etc.,  W^  will  re- 
ceive the  increment 

dW,   ,      .   dW,.      .   dW,  .    ,   .      , 

-dv/^'^  +  'dy;''^-^^''^-^''''^ 

and  VV^  will  be  increased  in  a  similar  manner  by  changing  x^ 
into  x^  -f-  dx^  and  varying  j^,  j^/o^  etc.  These  terms,  being  all 
suffixed,  cannot  control  the  general  solution,  which  must  be 
obtained  by  varying  V  in  the  usual  manner,  transforming  the 
variation  as  previously  explained,  and  solving  the  differential 
equation  J/  =  o  which  will  be  obtained.  Then  we  have  as 
the  terms  at  the  limits  those  derived  from  the  transformation 
of  dV,  together  with  those  derived  from  varying  W^  and  W^. 
Now  if  the  limits  be  fixed  we  shall  generally,  in  order  that  the 
number  of  limiting  equations  may  not  exceed  that  of  the  con- 
stants in  the  general  solution,  require  that  m  shall  not  exceed 
n—  I,  the  difficulty  in  the  last  problem  arising  from  the  fact 
that  m  is  equal  to  n.  But  if  the  limits  be  not  fixed,  we  shall 
also,  as  before,  require  usually  some  restriction  upon  the  ex- 
tremities of  the  curve  given  by  the  general  solution. 


Section  VI. 

relative  maxima  and  minima. 

Problem  XV. 

86.  It  is  required  to  find  among  all  plane  curves  of  a  given 
length  which  can  be  drawn  between  two  fixed  points,  that  wJiich, 
together  with  the  ordinates  of  its  extremities  and  the  axis  of  x, 
shall  contain  a  maximum  area. 

Whatever  be  the  nature  of  the  required  curve,  we  know 
that  its  length  is  y^     \/ 1  ^y^dx;  and  since  it  is  to  be  com- 


PLANE    CURVE    OF  MAXIMUM  AREA.  IO9 

pared  with  curves  of  the  same  length  only,  its  derived  curves 
must  not  differ  from  it  in  length,  and  we  must  therefore  have 

I     d  S^i-^ydx  —  o.     But  the  enclosed  area  is    /    ydA;;and 

since  this  is  to  be  a  maximum  for  all  changes  in  the  form  of 
the  curve  which  permit  its  length  to  remain  unaltered,  we 

must  have  also  to  the  first  order    /     dy  dx  =  o. 

Now  in  the  problems  hitherto  considered  no  restriction 
has  been  imposed  upon  the  variations  of  j/,  y,  etc.,  except  that 
they  must  always  be  infinitesimal,  and  the  curves  given  by  the 
general  solution  have  therefore  been  compared  with  all  others 
which  can  be  derived  from  them  by  such  variations.  The 
results,  therefore,  being  subject  to  no  restriction  so  far  as 
the  variations  are  concerned,  are  termed  absolute  maxima  and 
minima,  observing  that  the  terms  maxima  and  minima  are 
used  in  their  technical  sense  only,  and  not  in  that  of  greatest 
or  least.  But  in  the  present  problem  we  are  to  compare  the 
required  curve  with  such  only  as  can  be  derived  by  infini- 
tesimal variations  of  y'  without  any  increase  in  its  length, 
and  the  area  is  to  be  a  maximum  with  respect  to  such  varia- 
tions only.  That  is,  if  we  vary  the  required  curve  so  as  to 
increase  its  length,  the  area  need  no  longer  be  a  maximum. 
Examples  of  this  nature,  therefore,  are  termed  problems  of 
relative  jnaxima  and  ijiiniina,  and  also  isoperimetrical  problems, 
and  constitute  the  most  numerous  and  important  class  of  ques- 
tions discussed  in  the  calculus  of  variations. 


87.  Resuming  the  equations  of  the  last  article,  and  treat- 
ing the  first  as  usual,  recollecting  that  Sy^  and  Sy^  are  zero, 
we  have 

£yydx^O,  (2) 


no  CALCULUS  OF   VARIATLONS. 

which  signify  merely  that  any  values  of  Sy  which  will  satisfy 
(i)  must  also  satisfy  (2),  it  being  supposed  that  the  derived 
curve  has  been  obtained.  But  although  we  are  permitted  to 
pass  from  the  required  curve  to  such  derived  curves  only  as 
do  not  differ  from  it  in  length,  the  number  of  such  curves  may 
nevertheless  be  infinite,  so  that  we  cannot  express  in  an  ex- 
plicit form  the  nature  of  the  restriction  which  has  been  im- 
posed upon  6y,  or  rather  upon  Sy' ^  although  we  know  that 
such  variations  could  be  given  to  y'  as  would  not  satisfy  equa- 
tion (i),  and  might  or  might  not  satisfy  (2).  This  restriction 
prevents   us   from   employing  our  former  reasoning,  which 

d         y' 
would  here  2:ive  the  equations =  o,  the  differen- 

^  ^  dx  4/1  _^y2 

tial  equation  of  the  right  line,  as  appears  from  Prob.  I.,  and 

d         y 
the  impossible  equation  i  =  o.     Now  put  Z  for 


dx  4/1  _py^ 

Then  if  Z  can  be  a  constant,  it  is  evident  that  any  values  of  Sy 
or  dy'  which  will  satisfy  one  of  the  equations  at  the  beginning 
of  the  last  article  will  satisfy  the  other  also ;  and  we  will  now 
show  that  this  is  the  only  condition  which  will  insure  that  ^y 
cannot  be  so  taken  as  to  satisfy  one  equation  and  not  the 
other. 

88.  Let  x^,  x^,  x„  x^  be  four  particular  values  of  x  chosen 
as  hereafter  explained,  and  let  s  denote  the  value  of  the  inte- 


gral J  Sy 


dx  when  the  limits  are  ;ir„  and  x..  and  /  its  value 


when  the  Hmits  are  x^  and  ^..  Then  supposing  the  required 
curve  to  be  obtained,  let  us  make  Sy  zero,  except  from  x^  to 
x^,  and  from  x,  to  x, ;  that  is,  leave  the  required  curve  un- 
varied in  form  except  between  these  limits.  Also  let  us  give 
to  ^y  an  invariable  sign  from  x^  to  x^,  and  an  invariable  but 
contrary  sign  from  x^  to  x^.     Then  we  shall  have 

r'dydx  =  s  +  t.  (3) 


PLANE   CURVE    OF  MAXIMUM  AREA.  Ill 

Now  although  neither  s  nor  t  separately  vanishes,  we  can 
so  take  ^y  that  their  sum  shall  vanish,  and  thus  (i)  be  satisfied. 

Next  let  q  denote  the  value  of  the  integral   /  Zdydx  when  the 

limits  are  x\  and  x\,  and  r  its  value  when  the  limits  are  x^ 
and  x^.  Now  the  four  values  of  x  may  also  be  so  taken 
that  Z  will  be  of  invariable  sign  from  x^  to  x^,  and  also  from  x^ 
to  ^.,  it  being  of  no  importance  whether  the  signs  be  the  same 
or  not  for  these  two  intervals.  We  can  now,  with  the  values 
of  Sy  formerly  chosen,  secure  that,  unless  ^  be  a  constant,  q 
and  r  shall  be  numerically  unequal,  and  consequently  that 
their  sum  shall  not  vanish.     But,  as  before, 

r'Z6ydx  =  q  +  r,  (4) 

tyxo 

and  hence,  if  Z  be  variable,  we  can,  without  violating  the  re- 
striction which  has  been  put  upon  dy,  give  it  such  values  as 
will  satisfy  equation  (2)  but  not  (i),  which  is  contrary  to  the 
conditions  of  the  question. 

89.  Now   since  Z  is  a   constant,   let   it   equal  -.      Then 

a 

aZ  =  I ;  and  restoring  the  value  of  Z,  we  may  write 

d         y' 


\  —  a 


dx  4/1  _|_y 


=  O,  (5) 


an  equation  which  involves  the  coefficients  of  ^y  dx  in  both 
(i)  and  (2),  and  is  necessarily  true,  being  equivalent  to 
1  —  1=0.  But  it  will  be  seen  that  this  differential  equation, 
which  combines  both  conditions  of  the  question,  would  also 
have  been  obtained  if  we  had  at  first  required  to  maximize  or 
minimize  the  expression 


U=S^l\y-^a^iJry')dx, 


112  CALCULUS  OF    VARIATLONS. 

the  exti^eme  co-ordinates  being  fixed,  and  dy  or  Sy'  being 
subject  to  no  restriction.  Moreover,  we  shall  presently  show 
that  all  problems  of  this  sort  can  be  treated  in  a  similar 
manner. 

Now  integrating  (5),  we  have 


and  solving  for  y\  we  have 


Va  —  \c  —  X) 


Whence,  by  integration,  we  obtain 


y-^d=   \/a'-(c-x)\  (8) 

which  shows,  if  we  employ,  as  we  have,  the  positive  sign,  that 
the  required  curve  must  be  a  circular  arc,  in  which  a  must  be 
numerically  equal  to  the  radius  r. 

90.  Suppose  now,  as  just  suggested,  we  attempt  to  maxi- 
mize or  minimize  absolutely  the  expression 

Here  F  is  a  function  of  y  and  y' ,  and  P  =  ,  so  that 

y  I  +/" 

by  formula  (C),  Art.  56,  we  have 

ny 


y  +  aVi+y'  =  c  + 


Whence 


VT+Y'     '       '"     i^-yy 


PLANE    CURVE    OF  MAXIMUM  AREA.  II3 

which  must  be  solved  thus : 

Va'  -{c-  yf 

where  we  still  use  the  positive  sign.    Integrating  this  equation, 
we  have 

x  +  d=^Va'-{c-~yf,  (9) 

which  evidently  has  the  same  interpretation  as  before,  except 
that  c  and  ^need  not  be  identical  with  c  and  d  of  the  last  article. 

91,  It  will  be  seen  that  besides  the  two  constants  which 
arise  from  the  integration  of  (5),  which  we  may  call  AI  —  o, 
we  have  also  a  third  constant,  a.  But  now  we  also  have,  be- 
sides the  two  ordinary  conditions  given  by  assigning  the  values 
of  J,  and  Jo,  a  third  condition,  that  the  length  of  the  circular 
arc  shall  have  an  assigned  value ;  and  these  conditions  are 
sufficient  for  the  determination  of  the  three  constants. 

Consider  first  the  constant  a.  We  know  that  the  length 
of  the  chord  of  the  given  arc  is  V{x,  —  x,y  -\-  {y^  —  yo)\  and  is, 
therefore,  determined  as  soon  as  the  limiting  co-ordinates  are 
given ;  and  since  the  length  of  the  arc  is  assigned,  if  we  find 
an  expression  for  the  length  of  any  arc  in  terms  of  its  chord 
and  radius,  and  then  substitute  in  that  expression  the  known 
values  of  the  chord  and  arc  in  question,  we  can,  by  solving  for 
a,  determine  its  value  definitely.     This  expression  is 

{x,  -  x,Y  +  {y^  -  y^)-"  =  4r'  sin^  — ,  (lo) 

where  s  is  the  length  of  the  given  arc,  and  a  is  numerically 
equal  to  r,  its  sign  being  reserved  for  future  discussion. 

After  the  determination  of  a,  the  other  constants  are  readily 
found.     For,  from  the  general  equation  of  the  circle,  we  have 


y,+  d=±  Va'-ic-xy 


114  CALCULUS  OF  VARIATLONS. 

and  a  similar  equation  for  the  lower  limit ;  and  from  these  two 
equations,  when  the  sign  of  their  second  members  has  been 
agreed  upon,  c  and  d  can  evidently  be  expressed  in  terms  of 
a  and  the  given  limiting  co-ordinates. 

92.  We  will  now,  before  proceeding  further,  consider  the 
general  mode  of  treating  problems  of  relative  maxima  and 
minima. 

Suppose,  then,  we  require  that  /  vdx  shall  be  a  maxi- 
mum or  minimum,  v  being  any  function  of  x,  y,  y'.  .  .  .y^\ 
while  at  the  same  time  /  v^dx  is  to  remain  always  constant, 
v'  being  any  other  function  of  x,  y,  /,...  y^^\    Then  because 

vdx  is  to  be  a  maximum  or  minimum,  we  shall  have  to  the 


first  order 


t/Xo 


Xl 
dvdx  =  o;  (i) 


/*Xi 

and  because  /     v'dx  is  to  be  always  constant,  we  must  have 
absolutely 

r^dv'dx  =  o.  (2) 

Now  suppose  the  variations  of  these  integrals  to  be  found, 
and  transformed  by  integration  in  the  usual  manner.  Then 
if  we  make  Sy^,  (5>„,  d>/,  etc.,  zero,  we  shall  obtain,  from  (i) 
and  (2)  respectively,  results  of  the  form 

fySydx^O,  (3) 


£v'Sydx^O.  (4) 


PLANE   CURVE   OF  MAXIMUM  AREA.  "  II5 

But  Sy  being  restricted,  as  hitherto  explained,  we  cannot  say 

that  V  and  V  must  separately  vanish,  but  equations  (3)  and 

(4)  will  certainly  be  satisfied  simultaneously  if  we  can  be  sure 

that   V  is  always  equal  to   V  multiplied  by  some   constant ; 

V 
that  is,  that  — -  is  a  constant ;  and  we  will  now  show  that  no 

other  condition  will  satisfy  these  equations  simultaneously. 

93.  Supposing-  the  required  curve  to  have  been  obtained, 
choose,  as  before,  four  values  of  x  such  that  neither  Fnor  V 
shall  change  its  sign  while  x  lies  between  x^  and  x^  or  between 
x^  and  x^.  Now,  as  previously,  vary  the  form  of  the  curve 
between  these  two  intervals  only,  and  make  the  sign  of  dy 
invariable  for  each  interval  separately,  giving  to  it  the  same 
or  contrary  sign  for  these  two  intervals,  according  as  that  of 

Fis  contrary  or  the  same.     Then,  although  /  Vdydx  does  not 

vanish  when  taken  throughout  either  interval  separately,  we 
can  so  vary  y  as  to  make  the  integral  taken  throughout  the 
second  equal  to  the  same  integral  taken  throughout  the  first, 
but  with  a  contrary  sign.     But  we  have 

£  VSy  dx  =  /;'  VSy  d.r  +£''  V6y  dx,  (5) 

dy  being  zero  for  the  rest  of  the  curve.  Therefore  (2)  would 
in  this  case  be  satisfied.     Now  put  /  for  -— ,  then  (4)  will  be- 


come 


lyVSydx  =  o,-  (6) 

Sy  being  supposed  taken  as  before.  But  unless  /  be  a  con- 
stant, we  can  certainly  select  the  four  values  of  x  so  that  the 
two  integrals  m  the  second  member  of  (6)  shall  be  numeri- 
cally unequal,  in  which  case  their  sum  would  not  vanish  and 


Il6  '  CALCULUS   OF  VARIATLONS. 

(6),  or  rather  (4),  would  not  be  satisfied.  Hence /must  be  a 
constant  in  order  to  the  existence  of  a  relative  maximum  or 
minimum,  since  then  any  values  of  Sy  which  will  satisfy  (3) 
will  also  satisfy  (4),  while  otherwise  it  would  be  possible,  even 
from  among  the  restricted  values  of  (^J/,  to  select  such  as  would 
satisfy  one  of  these  equations  and  not  the  other. 

The  preceding  demonstration  is  due  to  Bertrand  (see  Tod- 
hunter's  History  of  Variations,  Art.  312,  and  also  the  seventh 
volume  of  Liouville's  Mathematical  Journal,  1842),  and  the 
author  most  heartily  agrees  with  Bertrand  in  regarding  the 
ordinary  method  of  treating  this  subject  as  insufficient. 

Now  write 

I  V 

then 

V+aV  =  V-  V=o. 

But  this  equation,  which  involves  V  and  V\  and,  being  true 
under  all  circumstances,  is  evidently  sufficient  for  the  solu- 
tion of  the  problem,  would  have  been  obtained  if  we  had  been 
seeking  to  render  U  an  absolute  maximum  or  minimum,  where 

U—         {z>-\~az'')d.v,  and  thus  Ave  are  enabled  to  substitute 

for  the  given  problem  a  problem  of  absolute  maximum  or 
minimum,  the  general  solution  of  which  will  be  identical  with 
that  which  we  require. 

This  method  is  due  to  the  illustrious  Euler,  who  first  re- 
duced the  treatment  of  this  class  of  problems  to  a  simple  yet 
comprehensive  rule.     (See  Jellett,  Introduction,  page  xvii.) 

It  is  evidently  immaterial  Avhich  of  the  quantities  v  and  7>' 
we  select  to  be  multiplied  by  a  constant.     For  if  we  have 

F— <7:F' =  o,  then  V^ -\- /^V— o,  where /^  = —.      Moreover,  we 

a 

may  also  give  the  constant  multiplier  any  form  which  may 

be  convenient,  as   —  a,  2a,   etc.,  its  value  being  ascertained 

subsequently. 


PLANE    CURVE    OF  MAXIMUM  AREA.  II7 

94.  Resuming  the  consideration  of  Prob.  XV.,  let  us  now 
examine  the  terms  of  the  second  order.  Here  a  difficulty 
presents  itself  in  the  outset  which  must  be  surmounted  before 
we  can  proceed.     We  find  that  the  variation  of  the  area  is 

simply  /     dydx,  there  being  no  additional  terms  of  the  second 

order ;  so  that  if  we  equate  this  variation  to  zero,  it  would 
seem  that  the  area  could  undergo  no  change  whatever  when 
the  curve  is  varied,  and  that  consequently  we  could  have 
neither  a  maximum  nor  a  minimum.  But  the  supposition  that 
the  terms  of  the  first  order  must  vanish  is  only  necessary 
when  there  are  terms  of  a  higher  order,  it  being  sufficient,  in 
a  case  like  the  present,  to  suppose  that  they  are  zero  so  far 
as  the  terms  of  the  first  order  are  concerned ;  that  is,  they 
need  not  be  zero  as  regards  Sy^,  Nevertheless,  as  we  cannot 
determine  the  nature  of  these  terms  of  the  second  order, 
should  any  exist,  we  shall  be  compelled  to  change  our 
method  of  investigation. 

Suppose,  then,  that  we  had  required  the  curve  of  mini- 
mum length  which,  together  with  its  extreme  ordinates  and 
the  axis  of  x^  shall  enclose  a  given  area.  Here  the  general 
solution  will  evidently  be  the  same  as  formerly.  For  pro- 
ceeding as  in  the  first  three  articles  of  this  section,  we  shall 
obtain  equations  identical  with  (i)  and  (2);  and  moreover,  by 
the  last  article,  we  see  that  by  Euler's  method  we  are  now 
merely  to  maximize  or  minimize  the  expression 


U=ll\^'^-\-y"-^by)dx, 


where  b  =  -.     But  the  enclosed  area,  instead  of  being  a  maxi- 


a 


mum,  IS  now  to  be  constant,  so  that   /     dydx  is  absolutely 

zero ;  while  the  length  of  the  required  curve,  instead  of  being 
constant,  is  now  to  become  a  minimum. 


Il8  CALCULUS  OF  VARIATIONS. 

It  should  here  also  be  noticed  that  while  the  length  of  the 
required  curve  was  to  be  constant,  equation  (i),  Art.  87,  can 
be  true  to  the  first  order  only.  For  since  the  variation  of  the 
length  contains  terms  of  an  order  higher  than  the  first,  and 
the  entire  series  is  to  vanish  absolutely,  it  is  clear  that  the 
term  of  the  first  order  must  equal  the  sum  of  the  others,  taken 
with  a  contrary  sign. 

As  the  area  gives  us  no  term  of  the  second  order,  we  have 
only  that  obtained  from  the  variation  of  the  required  curve, 
which  is 

^-0  2/(i+yy  -^  ^^ 


and  if  we  regard  Vi  +y'  as  positive,  the  length  of  the  curve 
is  evidently  a  minimum.  It  must,  however,  be  remembered 
that  Sy  and  dy'  are  restricted  to  such  values  only  as  will  satisfy 

the  equation  /     dydx  —  o.     But  since  (i)  is  positive  for  all 

real  values  of  Sy' ,  we  only  require  that  the  term  of  the  first 
order  in  the  variation  of  the  length  of  the  curve  shall  com- 

1       ,  .1  .  •    •  J     •  d      *  y' 

pletely  vanish  to  msure  a  mmimum ;  and  smce  — -^== 

^        ^  dx   \/i  _^ya 

is  a  constant,  this  condition  is  secured  when  we  make 
J     Sy  dx  vanish  absolutely. 

It  will  be  seen  that  equation  (i)  would  have  been  obtained 
had  we  found,  according  to  Euler's  method,  the  terms  of  the 
second  order  in 

b  being  -.  But,  as  before,  the  variations  are  not  entirely 
unrestricted,  since  they  can  have  such  values  only  as  will 
makey     dj^.r  vanish  absolutely. 


PLANE   CURVE   OF  MAXIMUM  AREA.  IIQ 

95.  Now  let  us,  according  to  Euler's  method,  consider  the 
problem  as  originally  given.     Then  we  shall  have 

U^ll\y  +  aVTT7^)dx.  (2) 


Here 

^ 

Hence 

dx  4/1  +y^ 

dx  ^ 

^(i+y?    '  '■' 

y 


(3) 


where  the  last  member  has  the  negative  or  positive  sign  ac- 
cording as  the  circular  arc  is  convex  or  concave  to  the  axis 

of  X.      Therefore  a  := =  ^t  r,  the  positive  or  negative 

sign  being  used  according  as  the  circular  arc  is  convex  or 
concave  to  the  axis  of  x.  Making  the  terms  of  the  first  order 
vanish,  (2)  will  give 

SU=  -  P    ^_  ""    _Sy'^dx=  ±  -   r^-  ^  _J d/V^.(4) 

2  ^^^   |/(i  -\^yy  2  ^^^   4/(1  ^y'y 

Hence  if  the  arc  be  concave  to  the  axis  of  x,  the  area  is  a 
maximum  ;  but  if  convex,  the  area  becomes  a  minimum ;  and 
these  results  are  evidently  as  they  should  be. 

It  must,  however,  be  remembered  that  we  have  not  as  yet 
shown  that  the  use  of  Euler's  method,  as  far  as  the  terms  of 
the  second  order,  must  in  this  latter  case  give  necessarily  a 
trustworthy  result,  but  merely  that  this  result  is  one  which  is 
known  to  be  true. 

96.  We  may  now  extend  the  discussion  of  this  problem,  and 
also  that  of  relative  maxima  and  minima  generally,  to  the  case 


120  CALCULUS  OF  VARIATIONS. 

in  which  the  limiting  values  of  x,  /,  y\  etc.,  are  also  subject  to 
change.     We  have  already  seen  that  if  we  seek  to  maximize 

or  minimize  an  integral  of  the  form  U  ==  J      Vdx,  the  general 

solution  will  be  the  same  whether  we  suppose  the  limiting 
values  of  x,  y,  y,  etc.,  to  be  fixed  or  not.  Hence  when  V  be- 
comes, as  it  will  in  the  use  of  Euler's  method,  v -}- av\  the 
general  solution,  obtained  under  the  supposition  that  the 
limits  are  variable,  will  be  identical  in  form  with  that  ob- 
tained by  supposing  those  limits  to  be  fixed.  Now  suppose 
we  add  to  Prob.  XV.  the  condition  that  the  required  curve 
shall  always  have  its  extremities  upon  two  given  curves;  and 
suppose  that  the  two  required  points  had  been  found  and  con- 
nected by  the  required  curve.  Then,  unless  this  curve  were 
a  circular  arc,  we  could  evidently,  from  our  preceding  dis- 
cussion, vary  it  so  as  to  increase  the  required  area  without 
changing  the  extremities  of  the  curve.  The  general  solution 
must  therefore,  as  formerly,  be  a  circular  arc,  the  only  ques- 
tion being  to  determine  the  position  of  its  extremities. 

The  reader  will  be  able  at  once  to  apply  a  similar  mode  of 
reasoning  to  any  problem  of  relative  maxima  or  minima  which 
may  present  itself;  and  therefore,  without  taking  space  to  gen- 
eralize the  demonstration,  we  shall  assume  that  the  general 
solution  of  these  problems  is,  like  those  of  absolute  maxima 
and  minima,  the  same  in  form  whether  the  limiting  values  of 
■^7  y,  y\  etc.,  be  fixed  or  variable.  Hence,  from  what  has  been 
said,  we  see  that  Euler's  method  may  be  employed  whether 
the  limits  of  integration  be  fixed  or  variable,  the  problem 
being  treated  in  all  respects  like  one  of  absolute  maxima  or 
minima. 

97.  Assume,  then,  in  order  to  discuss  the  limits, 

U^£^\y±r\/Y^Vy')dx.  (i) 


PLANE    CURVE   OF  MAXIMUM  AREA.  121 

If  we  suppose  first  that  x^  and  x^  are  fixed,  while  y^  and  y^  are 
variable — that  is,  that  the  arc  has  its  extremities  upon  the  two 
right  lines  whose  equations  are  x  =  x^  and  x  =  x^ — the  terms 
at  the  limits  evidently  become 

which  equations  signify  that  the  tangent  to  the  arc  at  each 
limit  must  be  parallel  to  x^  which  is  clearly  impossible.  But 
if  one  of  the  limiting  values  of  y  be  fixed,  the  tangent  at  the 
other  limit  can  be  drawn  as  described,  and  it  must  be  so 
drawn. 

Now  suppose  that  the  limiting  values  of  x  are  to  be  vari- 
able also.  Then  the  terms  at  the  limits  will  evidently  give  the 
equation 


(_,±,Vi+y^)_^,,±,j^_Z_.^tf, 


(2) 


with  a  similar  equation  at  the  lower  limit.  Let  the  extremities 
of  the  arc  be  confined  to  two  curves  whose  equations  are 
y  —  Fix)  ^  F,  y  —  f{x)  =  /.  Then  eliminating  Sy^  by  means 
of  equations  (2),  Art.  yS,  (2)  becomes,  after  omitting  the  com- 
mon factor  dx^, 

/  r  rfy'     \ 

and  a  similar  equation  for  the  lower  limit.  But  since  ds,  any 
element  of  the  arc,  equals  Vi  -\-  ydx,  (3)  may  be  written 


,   sm  n    . 

=  r,  ±  ^  ^  cos  m  + sm  ;// 

'  cos  n 


\ :  (4) 


122  CALCULUS  OF    VARLATIONS. 

where  m  is  the  angle  which  the  tangent  to  the  arc  makes  with 
the  axis  of  x,  and  n  the  angle  which  any  tangent  to  the  upper 
limiting  curve  makes  with  that  axis.  Let  t  be  the  angle  which 
the  tangents  to  the  arc  and  the  limiting  curve  make  with  each 
other  at  the  upper  limit.  Then,  since  t  is  numerically  equal  to 
n  —  m,  we  have 

cos  /j  —  cos  in^  cos  n^  -\-  sin  m^  sin  n^.  (5) 

Hence,  clearing  fractions,  (4)  gives 

r  cos  /j  =:  y^  cos  71^, 

and  we  can  establish  an  equation  of  a  similar  character  for  the 
lower  limit. 

It  must,  however,  be  remembered  that  none  of  these  results 
concerning  variable  limits  can  be  confirmed  as  true  maxima 
or  minima  without  an  examination  of  the  terms  of  the  second 
order,  which  examination  would  be  impracticable. 

Problem  XVI. 

98,  //  is  required  to  determine  the  form  of  the  solid  of  revolu- 
tion which  shall  possess  a  given  surface  and  a  maximum  volume, 
the  generating  curve  being  required  to  pass  through  two  fixed 
points  071  the  axis  of  revolution. 

Assume  x  as  the  axis  of  revolution.  Then  the  volume  to 
be  a  maximum  is   /     ny'dx,  while  the  given  superficial  area 

which  must  remain  constant  is  /     27ry  Vi  -\- y'"^  dx.     Hence, 

omitting  the  constant  tt,  we  have,  by  Euler's  method,  to  maxi- 
mize absolutely  the  expression 


MAXIMUM   SOLID   OF  REVOLUTION,  1 23 

Hence,  after  the  usual  transformation,  we  have 


dU 


+  r^  \  2v+2a  Vi  +y'  -  2a-^  — J^^  \  dydx,    (2) 

which  equation  is  evidently  true  whether  Sy^  and  Sy^  vanish  or 
not. 

Here,  as  Fis  a  function  of  r  and  /  onlv,  and  P  =  —  , 

we  have  by  formula  (C),  Art.  56, 

But  since  the  generating  curve  is  to  meet  the  axis  of  x,  c  must 
vanish,  and  we  have 


2   I         2^j/  /  2a       \ 


O. 


(4) 


Whence,  My  be  not  always  zero,  we  have 


,  2a 

Hence 

and 

which,  by  integration,  gives 

x  +  b=  ±   V^a'-f,  (8) 

the  equation  of  the  circle  whose  radius  is,  numerically  at  least, 


124  CALCULUS  OF  VARIATIONS. 

2a,  and  whose  centre  is  on  the  axis  of  x,  thus  rendering  the 
required  soHd  a  sphere. 

99.  We  are  evidently  prevented,  by  the  nature  of  this 
problem,  from  supposing  that  y  can  ever  become  negative, 
and  we  may  therefore  use  the  positive  sign  only  in  equation 
(8).  For  if  we  were  to  regard  y  as  negative  throughout  any 
interval,  say  from  x^  to  x^,  we  would  have  the  corresponding 
zone  of  surface  negative,  because  dx  and  Vi  -\^  y'^  are  taken 
positively,  which  would  be  absurd.  Hence  we  see  from  (5) 
that  2a  is  necessarily  negative ;  and  using  its  known  numerical 
value,  we  have  2<^  ==  —  r. 

100.  We  have  now  two  constants,  r  and  b,  to  determine, 
since  we  were  obliged  to  make  c  vanish  before  we  could  com- 
pletely integrate  equation  (3).  But  it  will  be  observed  that  it 
would  have  been  sufficient  for  a  solution  had  we  merely 
required  the  generating  curve  to  meet  the  axis  of  x  at  some 
point,  taking  this  point  as  one  of  the  limits,  say  the  lower,  and 
then  regarding  the  limits  as  variable.  By  this  method  we 
would  obtain  a  sphere,  as  before  ;  and  then  if  we  impose  the 
condition  that  both  extremities  of  the  generating  curve  shall 
be  confined  to  the  axis  of  x,  as  is  most  natural,  we  shall  have 
a  complete  sphere.  Hence,  since  the  superficial  area  is  given, 
r^  is  at  once  determined  by  dividing  the  area  by  ^.n,  and  the 
distance  x^  —  x^  being  necessarily  equal  to  2r,  becomes  also 
known  ;  so  that  when  one  limiting  value  of  x  is  given,  the  other 
can  be  readily  found.  Now  from  (8)  we  see  that  b  is  merely 
the  abscissa  of  the  centre  of  the  circle  or  the  sphere,  and 
equals  x^  -\-  r,  or  x^  —  r.  As  soon,  therefore,  as  one  of  the 
limiting  values  of  x  is  given,  all  the  required  quantities  can  be 
determined  ;  but  if  neither  x^  nor  x^  be  given,  r  only  can  be 
determined. 

(01.  Thus  far  there  would  seem  to  be  nothing  peculiar  or 
unsatisfactory  about  our  solution ;  but  we  come  now  to  speak 


MAXIMUM   SOLID    OF  REVOLUTION.  1 25 

of  a  point  which  has  occasioned  considerable  discussion  among 
mathematicians,  and  which  has  led  to  an  important  extension 
of  the  calculus  of  variations. 

Suppose  that,  as  in  the  original  enunciation  of  the  problem, 
we  require  that  x\  and  x^  shall  have  assigned  values,  or  that 
the  value  of  x^  —  x^  shall  be  assigned.  Then  the  diameter  of 
the  sphere  must  be  x^  —  x^,  and  the  only  value  which  the  sur- 
face of  such  a  sphere  can  have  is  7r{x^  —  x^y,  so  that,  as  we  are 
no  longer  at  liberty  to  select  a  value  for  the  superficial  area, 
the  solution  appears  at  first  to  fail.  But  it  has  now  been  made 
apparent  that  the  general  solution  of  any  problem  of  maxima 
or  minima  in  the  calculus  of  variations  is  entirely  independent 
of  any  conditions  which  may  be  required  to  hold  at  the  limits, 
the  limits  having  been  supposed  fixed  in  the  earlier  problems 
for  the  sake  of  simplicity  only.  Therefore  no  general  solution 
can  be  said  to  fail  so  long  as  it  is  always  possible  to  assume 
such  limiting  values  of  x,  y,  y\  etc.,  as  will  satisfy  all  the  con- 
ditions of  the  question  which  are  necessarily  involved  in  the 
general  solution. 

In  the  present  case,  if  we  require  that  the  surface  of  the 
solid  may  be  entirely  generated  by  the  revolving  curve,  these 
conditions  are  merely  that  the  value  of  the  superficial  area 
may  be  assigned  at  pleasure,  and  that  the  generating  curve 
shall  have  both  extremities  upon  the  axis  of  ;r,  which  condi- 
tions can,  as  we  have  seen,  always  be  fulfilled  by  a  sphere. 
Thus,  since  no  restriction  of  the  limits  x^  and  x^  is  implied  in 
the  method  by  which  the  general  solution  was  obtained,  the 
apparent  failure  of  the  solution,  when  these  limits  are  assigned, 
appears  to  arise  from  imposing  too  many  conditions  upon  the 
question,  some  of  which  are  incompatible,  and  for  this  the 
calculus  of  variations  is  evidently  not  responsible. 

It  will  be  remembered  that  in  Prob.  VII.  we  obtained  as  a 
general  solution  a  catenary,  having  its  directrix  upon  the  axis 
of  X,  and  then  subsequently  showed  that  the  two  fixed  points 
could  easily  be  so  taken  that  no  such  catenary  could  be  drawn. 


126  CALCULUS   OF  VARIATIONS. 

In  like  manner,  in  Prob.  XV.  we  shall  be  unable  to  draw  the 
required  arc  if  the  given  line  be  shorter  than  the  right  line 
which  joins  the  two  fixed  points,  or  longer  than  a  semicircum- 
ference  constructed  upon  this  right  line  as  a  diameter.  In  the 
first  of  these  problems  the  conditions  can,  without  changing 
the  limiting  values  of  x,  always  be  satisfied  by  assuming  suit- 
able values  for  y^  and  y^,  and  a  similar  remark  will  apply  to  the 
second  problem  unless  the  length  of  the  given  line  be  less  than 
x^  —  Xq,  in  which  case  some  change  will  become  necessary  in 
the  limiting  values  of  x  also. 

The  only  peculiarity,  then,  about  the  present  problem  would 
seem  to  be  that,  while  in  the  former  two  we  are  permitted  to 
make  various  but  not  all  possible  assumptions  regarding  the 
quantities  x^  —  x^  and  y^  —  y^y  here  but  one  supposition  regard- 
ing these  quantities  can  be  made  for  a  given  superficial  area, 
and  thus,  as  the  probability  of  failure  when  we  attempt  before- 
hand to  assign  the  limits,  and  also  the  surface,  is  vastl}^  greater 
in  this  problem  than  in  the  other  two,  it  more  readily  presents 
itself  to  our  minds. 

But  we  are  naturally  led  to  inquire  whether  there  may  not 
be  some  other  solution  for  this  and  similar  problems  in  those 
cases  in  which  the  general  solution  cannot  be  made  applica- 
ble. This  question,  which  has  received  much  attention  of  late, 
and  has  led  to  an  important  extension  in  the  calculus  of  varia- 
tions, will  be  discussed  in  a  subsequent  section  on  discontinu- 
ous solutions.  It  will  here  be  sufficient  to  say  that  such  solu- 
tions do  in  many  cases  exist,  and  are  generally  composed  of 
arcs  of  curves,  or  of  right  lines,  or  of  some  combination  of 
both,  and  they  are  hence  termed  discontinuous  solutions. 

(02  Now  if  we  put  for  2a  its  value  —  r,  the  general  equa- 
tion given  by  the  terms  at  the  limits  is 


^^y_ryV^+y'\d.r,-(--M^)lSy 


o,  (9) 


MAXIMUM   SOLID    OF  REVOLUTION.  12'] 

together  with  a  similar  equation  for  the  lower  Hmit ;  and  these 
equations  are  evidently  hke  those  of  the  preceding  problem, 
except  that  they  are  multiplied  by  y,  and  —  r  only  is  used.  If 
we  suppose  x^  and  x^  to  be  fixed,  and  j,  and  y^  to  be  variable, 
(9)  gives 


V^i+/7.    ""'       \^^+/-A 


o. 


Hence  we  may  have  y^  =  o,  y^  —  o,  thus  giving  an  entire 
sphere,  which  is  satisfactory  if  the  surface  Avill  permit.  If 
one  limiting  value  oi  y  be  also  given,  the  solution  can  always 
be  effected,  it  being  the  closed  segment  of  a  sphere,  having  a 
given  base  and  height,  r  being  determined  by  the  equation 


^._       4f_  (10) 

^  -  s  -  TtFf 

s  being  the  given  surface,  and  R  the  radius  of  the  base.  Re- 
garding the  other  solution,  j/  =  o,  j/;'  =  o,  it  may  be  remarked 
that  but  one  of  these  equations  can  ever  be  true,  and  therefore 
the  other  limit  must  be  fixed. 

Now  suppose  the  extremities  of  the  generating  curve  to 
be  limited  to  two  other  curves,  all  the  curves  being  supposed 
to  revolve  about  x,  which  is  the  same  thing  as  limiting  the 
sphere  to  two  surfaces  of  revolution.  Then,  since  the  terms 
at  the  limits  in  this  and  the  preceding  problem  compare  as 
we  have  just  shown,  it  will  appear,  by  methods  precisely  like 
those  employed  in  Art.  97,  that  we  shall  have 

ry^  cos  /,  =  y^  cos  7i^,  (i  i) 

together  with  a  similar  equation  for  the  lower  limit.  Thus 
we  have  either  ji  =  o  and  y^  =  o,  giving  a  complete  sphere, 
or  else  the  relation  given  in  the  last  equation  of  Art.  97. 

To  interpret  this  relation,  let  aj>  be  the  upper  limiting  curve. 


128  CALCULUS   OF    VARIATLONS. 

p  the  point  of  intersection  with  the  arc  whose  centre  is  c^py 
the  ordinate  y^  of  the  limiting  curve,  and  np  the  normal. 


Then   cpn  =  /„  and   npy  ~  n^,  and  we   have 

r  _  cos  n^ 
ji        cos  tl 


(12) 


and  this  equation  can  only  be  satisfied  by  making  cos  t^  unity, 
which  shows  that  the  tangent  to  the  limiting  curve  at/  must 
be  parallel  to  the  axis  of  x ;  that  is,  that  j,  must  be  either  a 
maximum  or  .minimum  ordinate.  But  if  y^  should  become 
equal  to  r,  this  relation  would  no  longer  be  necessary,  for 
then  the  lines  cp  and  yp  would  coincide,  the  angles  cpn 
and  ypn  become  the  same  angle  cpn^  and  (12)  becomes  merely 

T         COS  ci)7t 

-  = £_,  which  determines  nothing  regardins:  the  direction 

r      cos  cp7i 

of  the  normal  or  tangent  to  the  limiting  curve ;  and  hence  in 
this  case  the  ordinate  jj/^  need  be  neither  a  maximum  nor  a  mini- 
mum. 

103.  It  must  not,  however,  be  assumed  that  all  the  results 
obtained  in  the  last  two  articles  will  necessarily  render  the 
volume  either  a  maximum  or  a  minimum.  For  we  have 
already  seen  that  it  is  always  necessary  to  appeal  to  the  terms 
of  the  second  order  before  the  results  obtained  by  making 
those  of  the  first  vanish  can  be  interpreted.  We  have,  more- 
over, also  stated  that  the  discussion  of  these  limiting  terms, 
when  the  general  solution  is  known,  is  a  problem  of  the  differ- 
ential calculus  rather  than  of  the  calculus  of  variations,  and 
particularly  so  when  the  terms  of  the  second  order  are  to  be 
considered.     As  a  means  of  illustrating  both  these  remarks, 


MAXIMUM   SOLID   OF  REVOLUTION.  1 29 

we  shall  consider  only  the  case  in  which  one  limiting-  value  of 
y  is  zero,  and  take  the  liberty,  as  that  work  is  now  inaccessible 
to  most  readers,  of  copying  the  discussion  from.  Todhunter's 
History  of  Variations,  p.  408. 

Let  j^  be  any  ordinate  of  the  limiting  curve,  //  the  height  of 
the  segment,  v  its  volume,  and  s  its  surface.  Then,  since  the 
segment  is  known  from  the  general  solution  obtained  from 
variations  to  be  always  spherical  in  form,  and  by  supposition 
has  but  one  base,  we  have,  r  being  the  radius  of  the  sphere. 


(-■-?)■  (0 


7t  \rh 


and  we  can  now,  by  the  differential  calculus,  determine  the 
conditions  which  will  render  v  a  maximum  or  a  minimum,  sup- 
posing s  to  remain  constant.  Since  s  =  2nrJi  is  to  remain  con- 
stant, rh  is  a  constant,  say  k^.  Then  from  the  equation  of  the 
circle,  when  the  origin  is  at  the  extremity  of  any  diameter,  we 
have 

/  =  2rh  -  h'  =  2k''  -  le ; 
whence 

je  =  2k'-y\ 
and  therefore  (i)  becomes 


.  =  n\k^S/W^f-^(WEir^ 


(2) 


Whence 

and   since    the    differential   of   the   limiting    curve    must   be 
dy  =y^dx,  we  have 

dv_  _  7ry/(k'--f)  .  . 

dx  ^2k''  -  f  ' 


130  CALCULUS  OF    VARLATLONS. 

To  make  the  second  member  of  this  equation  vanish,  we  must 
have  y  ^  o,  y  =^  k,  or  y  =.  o. 

To  test  these  solutions,  write  u  =  k^  —  y^,  z  —  V2k''  —  y . 
Then 


dx 


J  =  J  {^'^^r  +  ^W  -  2zyy'^  +  uyY^.  (5) 


Whence  it  readily  appears  that  if/  vanish,  makings  a  maxi- 
mum or  a  minimum  ordinate  according  as  y"  is  negative  or 
positive,  V  will  have  the  like  or  contrary  property  to  y  ac- 
cording as  u  is  positive  or  negative. 

\i  y—  k,  without  making  y'  vanish — that  is,  without  being 

at  the  same  time  a  maximum  or  aminimum  ordinate — will 

be  negative,  and  7/  will  be  a  maximum.  But  if  y,  while  equal 
to  k,  be  also  a  maximum  or  a  minimum  ordinate — that  is,  make 

y  vanish —  — --  will  also  vanish,  and  it  will  be  found  by  trial 

74 

that  the  third  differential  will  do  so  likewise,  while  — -^  will 

dx    ' 

become  negative  or  positive  according  as  jj/  is  a  maximum  or  a 
minimum,  thus  making  v  have  the  like  maximum  or  minimum 
property  with  j. 

We  have  already  seen  that  the  question  does  not  permit 
us  to  suppose  that  J  can  become  negative,  and  hence  the  limit- 
ing curve  must  be  such  that  when  y  is  zero  it  shall  be  a 
minimum  ordinate,  which  will  cause  y'  to  become  zero  also. 

74 

These  suppositions  will  render  — i  positive,  having  reduced 

the  preceding  differential  coefficients  to  zero.  Therefore  the 
supposition  that  j/  is  zero  renders  v  a  minimum. 

The  foregoing  results,  which  have  been  verified  by  the 
author,  appear  to  be  correct,  although  they  do  not  agree  with 
those  obtained  by  Prof.  Jellett.     (See  his  page  165.) 


MAXIMUM  SOLID   OF  REVOLUTION.  I3I 

We  have  not  yet  examined  the  terms  of  the  second  order 
in  the  general  solution  obtained  by  the  calculus  of  variations 
in  the  problem,  as  originally  given,  but  shall  resume  this  point 
hereafter. 

104.  It  will  be  remembered  that  we  were  unable  to  in- 
tegrate equation  (3),  Art.  98  (that  is,  the  equation  M  =  o), 
without  supposing  c  to  become  zero.  Nevertheless  this  dif- 
ferential equation  has  been  shown  to  be  that  of  a  curve  traced 
by  the  focus  of  some  conic  section  as  it  is  rolled  along  the 
axis  of  X,  and  the  following  outline  of  the  demonstration  is, 
with  some  difference  of  notation,  given  by  Prof.  Jellett  on 
page  364,  but  the  proof  is  due  to  Delaunay. 

Let  r  =f{v)  =/  be  the  polar  equation  of  any  curve,  the 
pole  being  assumed  at  pleasure;  and  when  that  curve  is  rolled 
along  the  axis  of  x,  let  y  =  F{x)  =  F  hQ  the  equation  of  the 
curve  traced  by  that  pole.  Then  the  following  relations  are 
not  difficult  to  estabhsh  : 

By  means  of  these  relations  we  are  sometimes  able,  when  the 
equation  (differential  or  other)  of  one  curve  is  known,  to  deter- 
mine that  of  the  other ;  and  such  is  the  case  in  the  present 
instance.     Now  write  equation  (3),  Art.  98,  thus  : 


h  =  {f^d)^V^\ry^^  (3) 

where  b  =  —  2a  and  d——c.     Then,  from  (i),  we  obtain 


rdv 


32  CALCULUS   OF    VARLATLONS. 


Substituting  in  (3)  the  values  of  y  and   Vi  +y^  from  (2)  and 
(4),  we  obtain 

dv  =  _^£^_.  (5) 

r  ybr  —  r^  —  d 
The  integral  of  this  equation  is  known  to  give 


I         b        J  b'         I 

-  =  — 7  — r  —75  —  -7  cos  z; 

r       2d  4d'       d 


=z-^^^^°^^'  (6) 


where  A  =:  —.     If  now  we  assume,  as  the  polar  equation  of  the 

conic  section, 

I  _  I  -\-  c  cos  V 

r  ~    A{i  -e")  '  ^^^ 

we  can  obtain  from  it  equation  (6)  by  merely  making  e  equal 

/         d' 
to  y  I — ,  and  hence  the  truth  of  the  proposition  is  estab- 

A 

lished. 

The  curves  which  may  be  thus  described  are  exceedingly 
various.  Thus,  if  we  make  d  =:  —  c  vanish,  the  conic  section 
will  become  a  straight  line,  and  the  curve  generated  will  be  a 
circle,  giving  a  sphere  as  a  general  solution,  which  agrees 
with  what  has  been  already  shown.  Moreover,  the  circle  is 
evidently  the  only  one  of  these  curves  which  can  ever  meet 
the  axis  of  x.  Again,  if  we  take  the  circle  as  the  conic  sec- 
tion, the  curve,  traced  by  its  focus — that  is,  its  centre — will  be 
a  right  line  parallel  to  the  axis  of  x,  and  the  required  solid  will 
be  a  cylinder. 


CURVE   OF  LOWEST  CENTRE    OF  GRAVITY.  1 33 

Problem  XVII. 

105,  //  is  required  to  determine  the  form  which  a  uniform 
cord  of  given  lengthy  whose  extremities  are  confined  to  two  fixed 
points  or  curves^  must  assume  in  order  that  its  centre  of  gravity 
may  be  at  a  maximum  depth. 

Take  the  horizontal  as  the  axis  of  x,  and  let  L  or 

be  the  given  length  of  the  cord,  which,  by  the  conditions  of 
the  problem,  is  to  remain  constant.  Then,  by  the  well-known 
principles  of  finding  the  co-ordinates  of  the  centre  of  gravity 
of  any  curve,  we  shall  have,  D  being  the  depth,  which  is  to 
become  a  maximum, 

Hence,  by  Euler's  method,  we  are  to  maximize  absolutely  the, 
expression 

Here  Fis  a  function  of  j/  and  y'  only,  and 

p=    yy'  -^    "y'   ,  d) 

SO  that  by  formula  (C),  Art.  56,  we  have 


134  CALCULUS  OF  VARIATIONS. 

Whence,  by  reduction,  we  obtain 

y  -|-  aL 


and 


-  cL,  (3) 


which,  to  be  rendered  integrable,  must  be  solved  thus : 

Integrating  this  equation,  we  obtain 

x^  Al{y-\-  B  ^  ^{y  ^  By  -  A")  +  C,  (6) 

where  A  —  cL,  B  —  aL.  Comparing  this  equation  with  equa- 
tion (5),  Art.  59,  we  see  that  it  is  also  the  equation  of  a  cate- 
nary, in  which  y -^  B  \^  put  for  y ;  because  the  reasoning  in 
Art.  59  will  apply  equally  to  any  curve  whose  equation  is 
of  that  form,  and  this  equation  will  take  that  form  if,  while 
keeping  the  axis  of  x  horizontal,  we  remove  it  so  as  to  make 
B  zero.  Indeed,  without  integrating,  we  may  at  once  reach 
this  conclusion.  For  by  comparing  equation  (3)  with  equa- 
tion (2),  Art.  59,  we  see  it-  to  be  the  differential  equation  of  a 
catenary,  as  described. 

106.  To  determine  the  constants  A^B,  C,  we  have  the  con- 
ditions that  the  curve  must,  if  its  extremities  be  fixed,  pass 
through  those  fixed  points,  and  must  have  also  a  given  length, 
and  these  three  conditions  are  sufficient,  assuming  that  we 
can  solve  any  exponential  equation  which  may  arise.  Com- 
paring (6)  with  equation  (5),  Art.  59,  we  see  that  if  we  make 
the  axis  of  x  pass  through  one  of  the  given  points,  and  esti- 
mate y  upward,  B  will  be  the  distance  from  the  axis  of  x  to 


CURVE   OF  LOWEST  CENTRE    OF  GRAVITY.  1 35 

the  directrix,  estimated  positively  ;  but  if  we  estimate  y  down- 
ward, B  will  have  the  same  numerical  value,  but  will  be  nega- 
tive.    We  adopt,  however,  the  former  supposition.     Then,  as 

L  is  positive,  a  ox  —  must  be  also  taken  positively.    We  may, 

if  we  choose,  dispose  of  the  constant  C,  as  we  did  of  the  con- 
stant b  in  Art.  59,  by  making  \t  —  AlA. 

If,  then,  we  can  determine  A^  the  discussion  of  the  con- 
stants will  be  complete ;  and  this  may  be  done  in  the  follow- 
ing manner:  Let  D  denote  x^  —  x^,  which  is  supposed  to  be 
known,  and  £,  j\  —  jKo,  which  is  also  known,  and  let  the  ordi- 
nate, drawn  to  the  lowest  point  of  the  catenary,  divide  D  into 
two  segments,  /  and  ^,  while  the  corresponding  segments  of 
the  arc  L  are  m  and  n,  so  that  we  have 

/+^-A  '  (7) 

m-\-n  =  L.  (8) 

Then,  in  discussions  of  the  catenary,  the  following  equation 
is  easily  established  : 


7n 


=  ^{e'-e-^), 


(9) 


together  with  a  similar  equation  between  ^  and  Jt.     Whence 

/J  /  L       -L       L       -i\ 
L=-le^-e    ^J^e^-e     ^V  (10) 

Now  because   the   catenary  passes   through   the   two   fixed 
points,  we  have  from  its  equation,  (10)  of  Art.  59, 


A  /  L        -I        ^        -^\ 


(II) 


136  CALCULUS  OF  VARLATIONS. 

which  equation,  combined  with  the  preceding  four,  will  evi- 
dently determine  A,  which  in  statics  denotes  numerically  the 
tension  which  the  cord  will  sustain  at  its  lowest  point. 

107.  If  the  extremities  be  not  fixed,  but  merely  confined 
to  two  curves,  the  general  solution  will  of  course  be  un- 
changed,  only  certain  conditions  must  hold  at  the  limits.  For 
now  the  limiting  terms,  which  vanished  when  the  extremities 
were  fixed,  become 

V^dx,-  V,dx,  +  P,dy,-P,6y,  =  o.  (12) 

Substituting  the  values  of  P  and  V  from  (i)  and  the  preced- 
ing equation,  (12)  gives,  for  the  upper  limit, 

together  with  a  similar  equation  for  the  lower  limit.  Let  the 
equation  of  the  upper  limiting  curve  be  dj/=/'dx.  Then 
eliminating  (5>,  by  the  equation 


(13)  gives 


<y/,  =  (/' -/)y-*»  (14) 

(I +/.'//)  =  0.  (IS) 


Now,  to  make  the"  first  factor  vanish,  we  must  have  either 
y  =  —  aL  =  —  B  or  y  =  ±00.  But  since  B  is  numerically 
equal  to  the  distance  of  the  directrix  from  the  axis  of  x,  this 
supposition  would  make  the  lowest  point  of  the  cord  touch 
the  directrix,  and  this  could  not  be  unless  the  tension  were 
zero,  in  which  case  the  cord  would  hang  in  a  double  right 
fine,  having  its  extremities  at  a  common  point.     Neither  can 


CURVE   OF  LOWEST  CENTRE   OF  GRAVITY.  1 3/ 

we  suppose  j//  infinite.  For,  from  the  general  equation  of  the 
catenary,  we  have,  by  differentiation, 

y^\{^-c'^;  (i6) 

and  to  make  this  infinite  Ave   must  have  —  infinite,  giving 

either  A  zero,  which  condition  has  just  been  discussed,  or  x^ 
zero,  which  would  make  the  catenary  a  right  line  as  before. 

Hence  we  must  have  i  +7///  =  o,  and  a  similar  condition 
will  evidently  hold  if  the  other  extremity  be  confined  to 
another  hmiting  curve.  Therefore  we  conclude  that  the 
catenary  will  cut  its  limiting  curves  at  right  angles,  the  con- 
stants in  this  case  being  determined  by  the  conditions  that 
the  catenary  must  have  a  given  length,  and  that  its  extremi- 
ties must  cut  two  given  curves  at  right  angles,  or  pass 
through  a  fixed  point  and  cut  one  given  curve  at  right  angles. 

108.  The  terms  of  the  second  order  in  the  case  in  which 
the  extremities  are  fixed  are 

SU^-r—^^^^      .dy'^dx,  (i;) 

Now  from  equation  (3)  we  see  that  y-\-B  must  be  of  the 
same  sign  as  c.  Now  writing  {i)  y  -^  B  —  c  Vi  -{-/%  differen- 
tiating and  dividing  by  dy  or  /dx,  we  obtain 

and  therefore  c,  and  consequently  y  -f  B,  must  be  of  the  same 
sign  as  /'.  Now  if  we  estimate  y  upward,  the  catenary  is 
convex  to  the  axis  of  x,  and  /',  and  therefore  y  +  B,  is  posi- 
tive, and  we  have  a  minimum.  But  if  we  estimate  y  down- 
ward,  y-\-  B  is  negative,  and  we  have  a  maximum. 

These  different  results  appear  to  be  due  to  the  fact  that 


138 


CALCULUS  OF  VARLATIONS. 


by  estimating  y  upward  we  make  the  distance  of  the  centre 
of  gravity  from  the  axis  of  x  approach  as  near  as  possible  to 
—  00 ,  while  by  estimating  y  downward,  we  make  it  approach 
as  near  as  possible  to  +  ^  >  its  numerical  value  in  either  case 
being  the  same,  and  a  maximum. 

109.  If  we  assume  the  vertical  as  the  independent  vari- 
able, the  general  solution  must  be  the  same  whether  we  can 
obtain  it  by  that  method  or  not.  For  whatever  change  can 
be  made  in  the  form  of  the  required  curve  by  ascribing  varia- 
tions to  y  and  its  differential  coefficients  with  respect  to  x^ 
can,  at  least  if  the  curve  be  continuous  and  drawn  between 
fixed  points,  also  be  made  by  ascribing  suitable  variations  to  x 
and  its  differential  coefficients  with  respect  to  y,  y  itself  re- 
ceiving no  variation.  This  principle  will  be  found  to  aid  us 
in  the  solution  of  the  following  problem. 

Problem  XVIIJ.* 

110.  It  is  required  to  draw  between  two  fixed  points  A  and  B 
a  curve  of  given  length  having  the  following  property :  that  if  at 
any  point  S  of  the  required  curve  an  ordinate  NS  be  drawn,  and 
on  it  we  lay  off  NP  equal  to  the  arc  A  S,  the  curve  traced  by  the 
point  P  shall  enclose  a  maxiinum  or  minimum  area. 


N 


*  This  problem  is  only  a  particular  case  of  the  second  of  the  celebrated  iso- 
perimetrical  problems  given  by  James  Bernoulli,  the  original  problem  requiring 
NP  to  be  any  function  of  the  arc  AS,  which  can,  of  course,  not  be  fully  solved  so 
long  as  the  nature  of  the  function  is  entirely  undetermined.  The  solution  is 
from  the  Adams  Essay,  Chapter  XI. 


JAMES  BERNOULLFS  PROBLEM.  139 

Here  the  area  to  be  made  a  maximum  or  a  minimum  is 
I    ^sdxy  s  being  the  length  of  the  arc  measured  from  A,  while 

/  '  |/i  j^  ydx  (that  is,  the  length  of  the  arc  A  SB)  is  to  re- 
main constant.  Hence  we  are,  by  Euler's  method,  to  maxi- 
mize or  minimize  the  expression 

u=Z\^+^^^+yY--    ■        (I) 

Hence,  to  the  second  order, 

su=r'\6s-\-   /-^'^   _dyH — —^ sy' \  dx.   (2) 


But 


^=£y'+y'^^' 


Whence 

ds  ^  r    ^  ^        dy  dx  +  -  r      J^     —  dy  dx,       (3) 

and 

/ c^^  ^,r  =  xds  -f  X  -£  dx.  (4) 

Hence,  taking  this  integral  from  x^  to  x^,  and  observing  from 
the  figure  that  x^  is  zero,  while  b  may  be  put  for  x^,  because  it 
is  constant,  we  have 

rssdx^b  r\--jL==-6y4 ^  ^Syldx 

"^^  ^"   (  Vi+y  -^     2i/(i  +yy      ) 

-  f\\      y'    -dy-i ^       -dy^l^;r 

^0    ^  |/i_|_y^  -^  ^2i/(i  +y7      S 

=  f\t-x)\   ,  ^'     sy  +     ^^1_ dy^ X dx.       (5) 


140  CALCULUS   OF  VARLATIONS. 

Substituting  this  value  in  (2),  and  employing  the  usual  nota- 
tion for  the  limits,  we  have 

dU=  r\a  -^b-  x)\-    A Sy'  -\ -1 Sy'^  X  dx.  (6) 

Now  examining  the  second  factor  of  the  second  member  of 
(6),  we  see  that  it  is  the  variation  of  Vi  +y  dx,  or  ds,  and  that 
b  —  X,  or  Z,  is  the  distance  of  any  point  of  the  arc  ASB  from 
the  line  BF,  and  therefore  it  is  not  difficult  to  see  that  the  prob- 
lem is  really  in  solution  as  though,  taking  the  vertical  as  the 
independent  variable,  we  had  required  the  form  of  the  curve 
of  given  length  passing  through  A  and  B,  and  having  the  dis- 
tance of  its  centre  of  gravity  from  BF  a  maximum  or  a  mini- 
mum. Therefore,  without  solving  (6)  in  detail,  we  know  from 
the  last  article  of  the  preceding  problem  that  this  curve  is  a 
catenary,  having  its  directrix  parallel  to  the  axis  of  y. 

in.  But  some  investigation  will  be  necessary  in  order  to 
determine  the  sign  of  the  terms  of  the  second  order.  For 
although,  as  before,  it  is  evident  that  a,  like  B  of  the  last  prob- 
lem, is  numerically  equal  to  the  perpendicular  distance  from 
BF  to  the  directrix,  its  sign  is  not  at  once  clear.  Treating 
the  terms  of  the  first  order  in  (6)  in  the  usual  way,  we  obtain 


{a^b- x)--l=._^c,  (7) 


Whence 


,+,_.  =  ii:i+Z".  (8) 

y 

Differentiating  (8),  and  dividing  by  dx,  we  have 


cf 


ywi+y 


=  h  (9) 


SOLID   OF  MAXIMUM  ATTRACTION.  I4I 

from  which  it  appears  that  c  must  always  be  of  the  same  sign 
as  y" .  But  the  catenary  may  be  either  convex  or  concave  to 
the  axis  of  x,  so  that  c  will  be  positive  in  the  former  and 
negative  in  the  latter  case.  Moreover,  we  see  from  (7)  that 
a-\-b  —  X  must  always  be  of  the  same  sign  as  c,  and  therefore 
the  terms  of  the  second  order  will  become  positive  when  the 
catenary  is  convex  to  the  axis  of  x,  and  negative  when  it  is 
concave,  thus  showing  that  the  area  in  question  will  be  a 
minimum  in  the  former  and  a  maximum  in  the  latter  case. 


Problem  XIX. 

112.  //  is  required  to  determine  the  form  of  the  solid  of  revo- 
lution of  given  mass  and  iinifor^n  density  which  ivill  exert  a  maxi- 
mum attractive  force  upon  a  particle  situated  upon  the  axis  of 
revolution. 

Take  the  axis  of  revolution  as  that  of  x,  and  let  the  at- 
tracted particle  be  at  the  origin.  Moreover,  conceive  the 
solid  to  be  divided  into  shces  of  the  thickness  dx,  by  planes 
perpendicular  to  x.  Then,  by  dividing  these  sHces  into  dif- 
ferential rings,  it  is  easily  found  that,  omitting  the  factor  of 
density,  because  constant,  the  force  exerted  by  any  slice  in 
the  direction  of  x  is 


27r  ;  I ,   ^         \  dx. 


Hence 


is  to  be  a  maximum,  while  the  volume  /     ny'^dx  is  to  remain 


142  CALCULUS  OF  VARIATIONS. 

constant.      Hence,  by  Euler's  method,  we  maximize  the  ex- 
pression 

U=  r'\i ^J. +  a/\dx  =  r'Vdx.  (i) 

Therefore,  to  the  second  order,  we  have 

6U=  r  \  ^-^^-3  +  2ay  \  Sydx 
.r(.r'  —  2/) 


+rl"-^ 


2(x^-\-yy 


Sfdx.  (2) 


Here  F  is  a  function  of  x  and  y  only,  and  the  terms  of  the 
first  order  in  (^^need  no  transformation;  so  that  we  have  at 
once,  unless  y  be  always  zero, 

^    ^  +  2^==o.  (3) 


{x^+/) 

Now  putting for  2a,  (3)  gives 

(^^+y)i^,^^.  (4) 

But  V,  the  volume,  or   /     rty'^dx,  is  a  known  quantity,   and 

(4)  gives 

y  =  (,V)§_^^  (5) 

Whence 

v=7t  r\c^  x^  -  x')  dx,  (6) 

But  the  general  integral  of  the  second  member  of  (6)  is 

;r(l.J^S-ly)+«?.  (7) 


SOLID   OF  MAXIMUM  ATTRACTION.  143 

Now  suppose  x^  to  be  zero,  which  will  place  the  particle 
upon  the  surface  of  the  solid ;  and  assume  also  that  when 
;r  ==  jTj  the  generating  curve  meets  the  axis  of  x.  Then,  by 
making  y  zero,  and  x,  x^,  in  (4),  we  see  that  x^  is  numerically 
equal  to  c ;  and  taking  (7)  between  the  limits  o  and  c,  we  find 

v  =  ^,  (8) 

15 

It  therefore  appears  that,  when  the  volume  is  given,  the 
length  of  the  axis  is  not  in  our  power,  but  is  determined  by 
that  volume  ;  and  c^  being  known,  a  is  also  known. 

113.  Now  the  coefficient  of  Sy"  dx  in  (2)  is 

Putting  for  a  its  value ^,  and  substituting  for  the  first 

members  of  (4)  and  (5)  the  second  members  of  the  same  equa- 
tions, (9)  becomes 

I        xf^ix"  -  2c^  x^)  I         3.^-t  -  2c% 

2C'^  2{ex)l  '  20"^  2C^        ' 

Hence  the  terms  of  the  second  order  become 

Now,  since  v  cannot  be  negative,  we  see  from  (8)  that  c  must 
be  positive,  and  it  is  numerically  greater  than  x^  being  equal 
to  x^.  Therefore  E  is  positive,  while  Z  can  never  become 
positive,  and  the  terms  of  the  second  order  become  invariably 
negative,  thus  giving  us  a  maximum. 


144  CALCULUS  OF  VARLATIONS. 

Problem  XX.* 

114.  It  is  required  to  determine  the  form  of  the  solid  of  revo- 
lution, having  a  given  base  and  volume,  which  will  experience  a 
mijii^num  resistance  in  passing  through  a  fluid  in  the  direction  of 
the  axis  of  revolution. 

Let  X  be  the  axis  of  revolution.     Then,  reasoning  as  in 

Prob.  VI.,  we  see  that  we  must  minimize  /       -^'-^   ,  dx,  while, 

the   volume   being  given,   we   must   have    /    y^dx  constant. 

Therefore,  by  Euler's   method,   we  minimize  absolutely  the 
expression 

Here  Fis  a  function  oi  y  andy,  and 

so  that  by  formula  (C),  Art.  56,  we  have 

We  will  assume  that  the  generating  curve  cuts  the  axis  of 
X,  which  will  render  b  zero,  and  then  we  easily  obtain  the 
equations 

-y  =  (T^=     ^"d  >  =  -^„         (4) 

c  being  put  for  -. 
a 

*  The  following  discussion,  which  is  much  more  satisfactory  than  that  of  Prob. 
VI.,  appears  to  be  almost  entirely  due  to  Prof.  Todhunter.  (See  his  Adams 
Essay,  Chapter  X.,  from  which  this  solution  is  taken.) 


PROB.    VI.    WITH  GIVEN  BASE  AND   VOLUME.  I45 

115.  Now  the  last  equation  can  be  shown  to  indicate  that 
the  generating  curve  is  a  hypocycloid.  For  let  y'  =  tan  v. 
Then,  by  (4),  we  have 

y  =  c  sin^  v  cos  v^  (5) 

and,  by  differentiation,  we  have 

y  =  c(^  sin^  V  cos^  V  —  sin*  v)  — 

dx 

di) 
=  ^(3  cos''  V  —  sin''  v)  sin^  v  -7-.  (6) 

Hence 

dy 

-^  =  ^(3  cos''  V  —  sin*  V)  sin*  v,  (7) 

Dividing  (7)  by  y'  =  tan  z'  = ^-,  we  have 

cos  V 

dx 

— -  =  ^(3  cos'  V  —  sin*  z))  sin  v  cos  ^.  (8) 

Squaring  and  adding  (7)  and  (8),  observing  that 

sin*  V  -f-  sin*  v  cos*  v  =  sin*  7/  (sin*  v  -\-  cos*  ^)  =  sin*  ^.i, 

we  have,  putting  ds  for  an  element  of  the  arc, 

ds 

—  =  ^(3  cos*  V  —  sin*  v)  sin  ^  r=r  ^  sin  ^v.  (9) 

To  integrate,  write  this  equation  thus : 

c 
ds  =  —  sin  3^/^(3^/). 

Then  we  obtain 

s=  -  ^cossv  +  c,.  (10) 


14^  CALCULUS  OF  VARIATIONS. 

This  equation  is  known  to  indicate  that  the  curve  is  a  hypo- 
cycloid,  the  radius  of  the  rolHng  circle  being  one  third  that 
of  the  fixed  circle.  If  now  we  suppose  that  when  y  van- 
ishes V  vanishes  also,  and  measure  s  from  this  point,  we  have 

o  = cos o-\-  c^\  that  is, <:=:  —  ;  and  (lo)  becomes 

3  3 

^  =  - (i  —  COS37/).  (11) 

116:  To  determine  the  constant  c,  we  have  the  conditions 
that  the  solid  must  have  a  given  base  and  an  assigned  volume, 
and  we  may  use  these  conditions  thus :  Let  v^  be  what  v  be- 
comes when  X  ^=^  x^  and  when  j/  =:j|/j,  a  known  constant,  say  B, 
Then  it  is  shown  that  the  volume  of  the  sohd  is 

TtB'i^- sin^  v,A sin'  v\  ,     , 

V8        10  '^3  V  (12) 


We  have  also,  from  (5), 

B  ^^  c^\vl  v^CQS>v^\  (13) 

and  from  these  equations  v^  and  c  may  be  determined. 

This  solution,  however,  like  some  others,  is  not  always 
possible.  For  it  is  shown  that  the  volume  can  be  as  great  as 
we  please,  but  that  it  diminishes  as  v^  increases,  and  has  its 

least   value   when   v^  =  — ,  its  value  then  being — •      If, 

therefore,  the  given  volume  be  less  than  this  quantity,  no  such 
solid,  with  the  given  volume,  could  be  constructed  upon  the 
given  base. 

117.  Let  us  now  examine  the  terms  of  the  second  order. 
We  may  evidently  write  1/  thus  : 

U=£\.yf+2af)dx,  (1-4) 


PROB.   VI.    WITH  GIVEN  BASE  AND   VOLUME.  1 47 

where  /=: — - — ^,  and   is   therefore   a  function  of  y'  only. 

■^     1 4-y'  •' 

Hence  the  terms  of  the  second  order  arising  from  the  expres- 
sion  /    yfdx  may  be  trea1:ed  as  in  Prob.  VI 1 1.,  while  the  term 

arising  from    /     2ay''dx  is  evidently  -  /     \aSy''dx.    Therefore, 

by  the  formula  of  Prob.  VIII.,  we  have,  Avhen  we  suppose  the 
limits  to  be  fixed, 


where 


and 


# 


f,^^^  3/1+/' 

^      dy'     (I  ^-y'r 


(16) 


But,  from  the  first  equation  (4),  we  have 

vH-iv"  —  1/'*') 

V'=^  (l^yy  -'         ^^hence    2a  =/"/'.  (18) 

Substituting  this  value  in  (15),  we  have 

^^=  \S,J\y^y''+y"^f)  dx.  (19) 

Now  f"  is  positive  so  long  as  y  does  not  exceed  three ;  that 

is,  when  v  does  not  exceed  -;  and 

3 
that  the  integral  becomes  positive. 


is,  when  v  does  not  exceed  -;  and  y"  is  here  positive  also,  so 


148  CALCULUS  OF  VARIATIONS. 

118.  But  since  the  distance  x^  —  x^  is  not  fixed,  it  is  evident 
that  the  limits  of  integration  are  not  altogether  fixed.  But  as 
the  base  is  given,  and  we  may  consider  its  distance  from  the 
origin  as  fixed,  the  Hmit  x^  may  be  regarded  as  fixed,  as  is  also 
j/j.  Now  the  terms  of  the  first  and  second  order  arising 
from  the  variation  of  x^  and  }\  evidently  are 

-  r.  dx,  -  Pfiy^  -  i  ^dx:  -  S  V,  dx,  -  L//  Sy:,      (20) 

the  last  term  resulting  from  the  formula  in  Prob.  VIII. ,  when 
dy  at  either  limit  does  not  vanish.     But 

P.  =  yJ^.       -j--  =  7o  /o'  y"  +/o  Jo'  +  4^jo/oi 

dx, 

^K  =  yj:  ^y:  +/o  ^y.  +  A^y.  ^jv 

Now  since  y^  is  zero,  and,  as  appears  from  (4),  j/  is  also  zero, 
all  the  quantities  V^,  f^,  //,  -— -  will  separately  vanish,  and  the 

UXq 

terms  in  (20)  will  disappear.  Therefore  the  variation  arising 
from  a  change  in  x^  and  y^  is  not  even  of  the  second  order, 
although  it  might  still  be  a  quantity  of  the  third  order;  and 
as  the  integral  in  (19)  is  positive,  we  have  in  this  case  a  solid 
of  minimum  resistance. 


Problem  XXI. 

(19.  Let  a  curve  meet  the  axis  of  x  at  tzvo  fixed  points,  the 
origin  being  assumed  midway  between  them.  Then  it  is  required 
to  determine  the  form  of  this  curve,  so  that,  being  revolved  about 
the  axis  of  x,  it  may  generate  a  solid  of  given  volume  whose 
moment  of  inertia,  with  respect  to  the  axis  of  y,  may  be  a  mini- 
mum. 


SOLID   OF  MINIMUM  MOMENT  OF  INERTIA.  1 49 

Conceive  the  solid  to  be  divided  into  slices  by  planes  per- 
pendicular to  the  axis  of  x.  Then  the  moment  of  inertia  of 
any  slice,  whose  thickness  is  dx,  is 


7tin  I  — 


'(-+xy\dx,  (,) 


where  m  denotes  the  mass,  which  is  constant.  This  equation 
is  easily  obtained  from  the  moment  of  inertia  of  the  rings  of 
which  the  slice  is  composed,  which  is 

m'^+-A  (2) 

J/ being  the  mass  of  the  ring,  or  27tmdydx.  Therefore,  since 
the  volume  is  to  remain  constant,  we  must,  by  Euler's  method, 
minimize  the  expression 

^'  -X?  1  f + -y  -  -y  \  ^- = X?  vdx.      (3) 

Of  course  we  could  have  put  a  for  —  a^  as  the  indeterminate 
multiplier,  and  this  is  what  we  would  naturally  do  in  first  in- 
vestigating the  problem  ;  still  the  present  form  is  known  to  be 
more  convenient. 

Now  we  have 

^^=  £yy^+  ^^'-^  -  ^^^^-^  ^-^  =Xy^y^-^  2'^'-  ^"^yy  ^^'  (4) 

Hence,  if  y  be  not  always  zero,  we  have 

/  +  2.r^==2^^  (5) 

which  shows  that  the  solid  must  be  an  oblate  spheroid  m 
which  the  major  axis  is  to  the  minor  as  V2  is  to  unity. 


ISO  CALCULUS  OF  VARIATIONS. 

120.  The  terms  of  the  second  order  are 

which,  by  means  of  (5),  reduce  to  SU  =    /    y'Sy'dx,  and    this 

being-  necessarily  positive,  we  have  a  minimum. 

But  while  the  solution  is  thus  apparently  satisfactory,  it 
evidently  affords  another  example  of  the  kind  discussed  in 
Prob.  XVL  For  if  we  suppose  the  limits  x^  and  x^  to  be 
assigned — that  is,  the  minor  axis  of  the  ellipse — then,  unless 

the  volume  be  just  ,  in  which  B  is  the  semi-minor  axis, 

no  such  spheroid  can  be  constructed.  But  if,  without  assign- 
ing the  limits  except  to  make  the  curve  meet  the  axis  of  x  at 
two  points  equally  distant  from  the  origin,  we  only  require 
the  figure  into  which  a  given  volume  must  be  formed,  as  above, 
we  shall  obtain  a  spheroid  in  which  the  axes  are  related  as  just 
mentioned,  the  limiting  values  of  x  having  been  determined 
by  the  given  volume.  Still,  in  the  investigation  of  the  terms 
of  the  second  order  just  given,  we  have  assumed  that  x^  and 
x^  undergo  no  change.  Nevertheless,  if  we  vary  x  and  y  at  the 
limits,  we  shall  not  increase  these  terms,  since,  y  at  the  limits 

being  zero,  F^,  V^,SV^,dV^A-j-\     and   (-^j  severally  vanish. 

Here  the  constants  are  all  determined  by  the  assigned 
volume,  combined  with  the  conditions  that  y^  and  y^  shall  be 
zero.    For  B  is  determined  from  the  condition  that  the  volume 

must  equal  an  assigned  quantity;  then  A,  the  semi-major 

3 
axis,  by  the  known  relation  between  the  axes ;  after  which  a"  is 

found  by  means  of  (5). 


PROBLEM   OF  LEAST  ACTLON.  I5I 


Section  VII. 


CASE  LN  WHICH  V  IS  A  FUNCTION  OF  POLAR    CO-ORDINATES 
AND    THEIR  DIFFERENTIAL   COEFFICIENTS. 

121.  The  principles  of  the  calculus  of  variations  thus  far 
obtained  are  equally  applicable  when  polar  co-ordinates  are 
to  be  employed ;  and  as  the  mode  of  applying  these  principles 
is  precisely  similar  to  that  which  we  have  already  given  for 
rectangular  co-ordinates,  we  shall  present  but  two  examples, 
the  first  of  absolute,  and  the  second  of  relative  maxima  and 
minima. 

Problem  XXII. 

A  particle  which  is  always  attracted  towards  a  fixed  centre^ 
with  a  force  which  varies  according  to  the  Newtonian  law  of 
gravity^  is  projected  from  a  fixed  point  so  as  to  pass  through  a 
second  fixed  point.  It  is  required  to  determine  the  nature  cf  its 
pathy  assuming  that  it  must  be  the  path  of  least  or  minimum 
action. 

Assume  the  attracting  centre  as  the  pole,  r  as  the  radius 
vector,  or  distance  of  the  particle  at  any  time,  from  the  centre 
of  force,  r^  and  r,  the  distance  of  the  first  and  second  points 
respectively,  and  v  the  natural  angle  included  between  r^  and 
any  other  radius  vector.  Also  let  /,  a  constant,  be  the  inten- 
sity of  the  force  at  a  unit's  distance,  and  v'  the  velocity  of  the 
particle  in  its  orbit  at  any  instant. 

Now,  from  mechanics,  the  action  of  the  path  is 


t/So 


'ds,  (I) 

where  ds  is  an  element  of  the   path.     But 

ds  =  Vdr'+Vdzr'  =  dvV  r'  +  $^'  =   Vr^+VV^,  (2) 

dv 


152  CALCULUS   OF  VARIATIONS. 

SO  that  the  action  becomes 

£\J  .fTTj^dv.  (3) 

Now  in  determining  v'  three  cases  arise.  For  we  know 
that  the  path  of  a  revolving  particle  will  be  an  eUipse,  a  parab- 
ola, or  an  h3^perbola,  according  as  v^,  the  velocity  of  projec- 
tion, is  less,  equal  to,  or  greater  than  y  — .    Let  us  here  con- 


sider the  first  case,  and  suppose  v'  —  y  —  —  --.     Then  it  is 

r,        a 

known  that  v'  will  equal 


/^~^.  (4) 


r        a 


Substituting  this  value  of  v'  in  (3),  and  omitting  the  constant 
/,  we  have  to  minimize  absolutely  the  expression 


^fyVT+T-'d.^fydv.  (5) 

Now  change  r  into  r  -\-  Sr,  and  r'  into  r'  -\-  Sr',  while  v  re- 
mains unvaried.  Then  we  can  develop  the  new  state  of  U 
just  as  we  could  if  in  U  we  had  put  x  for  v,  y  for  r,  and  y' 
for  r' .     Hence,  to  the  first  order,  we  have 


PROBLEM  OF  LEAST  ACTION.  1 53 

But,  as  in  plane  co-ordinates,  dr'  =  --— ,  so  that  dU  may  be 

dv 

transformed  in  the  usual  manner  by  integration  by  parts,  Sr^^ 
and  Sr^  vanishing  because  the  two  radii  are  fixed.  But  we 
need  not  perform  this  transformation,  which  would  give  an 
expression  not  readily  integrable.  For  the  formulas  of  Art. 
56  become  at  once  applicable  to  polar  co-ordinates  when  in 
those  formulae  we  substitute  v,  r,  r\  r\  etc.,  for  x,  y,  y\y\  etc. 
Here,  then,  F  is  a  function  of  r  and  r\  and 

dV  Wr' 

P     o^     -rr  —  —  ,  (n\ 

dr  4/^2_j_^/2  v// 

so  that  by  formula  (C),  Art.  56,  we  have 

Wr""       ,  ,  Wr' 


W  Vr'-{-  r"'  =      __!__.  +  c,         and     '' '      =  c,      (8) 

Vr'-\-  r'  Vr'+  r'' 


Solving  for  r^',  we  obtain 


r    = 


(9) 


where  3  =  c\     Now  put  -  for  r.     Then  the  following  equa- 


tions  will  be  found  to  hold  true 

: 

W'=2U-  -, 

a 

r'': 

I  dtl' 

ti'dv"' 

and  (9)  gives 

dll^  _2U 

dv'       b 

I 

-u\ 

(10) 


Solving  and  putting  C  for  — ,  we  have 

ab 

J  du 

dv  = 


b 


154  CALCULUS  OF    VARIATIONS. 

where  the  negative  sign  is  used,  because  --  = — .     Now 

dv  r  dv 

by  placing  75—75  within  the  radical  sign  in  (11),  that  equa- 
tion may  evidently  be  written  thus : 

,  —  du  dX 

dv  = 


^t-'^) -{«-,-)■  ""-"'    ""' 


Whence,  by  integration,  we  obtain 

I 

I  -1  ^  -\  b 

V  -\-  g—  cos      -7,-  =  cos 
K 


Whence 


and 


^  b' 


I 

^  —  7- 

COS(z/+^)r^  .  (13) 

^    b' 


u     or     i  =  ^+r  ^— ^'cos  (^  +  ^).  (14) 


Now  write  b  =  a{i  —  /),  and  replace  C  by  its  value  --.    Then 

c 
the  quantitv  under  the  radical  readily  reduces  to  —. --, 

^  -^  -^  a{i  —  e") 

and  we  have 

I  _  i+^cos(2;+^) 

Now  in  equation  (8),  in  order  that  c,  or  Vb,  may  be  a  real 
quantity,  we  must,  since  a  is  by  supposition  positive,  have 


PROBLEM  OF  LEAST  ACTION.  155 

I  _  e"  positive.     That  is,  e  must  be  less  than  unity,  and  (15)  is 
therefore  the  equation  of  an  ellipse. 

122.  It  appears  as  though  the  general  solution  contained 
four  arbitrary  constants ;  but  as  e  depends  upon  the  ratio  of  a 
and  by  the  semi-major  and  minor  axes,  the  number  of  arbitrary 
constants  is  only  three.  But,  as  in  former  examples,  the  gen- 
eral solution  is  totally  independent  of  the  possibility  of  render- 
ing it  appHcable  in  any  particular  case.  Of  these  constants, 
a,  or  the  semi-major  axis,  is  determined  as  soon  as/,  r^  and  < 
are  given,  but  must  of  course  be  of  sufficient  value  to  enable 
the  ellipse  to  pass  through  the  second  fixed  point.  The  least 
value  of  a  which  will  render  the  solution  possible  in  any  case 
may  be  determined  thus :  Since  the  distance  of  the  two  fixed 
points  from  the  first  focus  are  respectively  r^  and  r^,  their  re- 
spective distances  from  the  second  focus  must  be  2a  —  r^  and 
2a  —  r^.  Now  from  the  first  fixed  point,  with  a  radius  2a  —  r^, 
and  from  the  second,  with  a  radius  2a  —  r^,  describe  circular 
arcs.  Then  if  these  arcs  do  not  touch  there  can  be  no  solu- 
tion, the  least  admissible  value  of  a  being  that  which  will  cause 
them  to  touch,  while  if  a  be  increased  beyond  this  value,  the 
circles  will  cut,  and  there  will  be  two  positions  for  the  second 
focus ,  that  is,  two  ellipses  can  be  described  as  required. 

Thus,  although  we  seem  to  have  three  conditions  for  the 
determination  of  the  three  constants — namely,  the  intensity  of 
the  initial  velocity  and  the  distance  of  each  of  the  two  fixed 
points  from  the  focus — we  can  in  fact  only  determine  a.  This 
result  might,  however,  have  been  anticipated,  as  we  know 
from  mechanics  that  while  the  form  of  the  curve  and  the 
value  of  its  major  axis  depend  solely  upon  the  values  of  /, 
v^'  and  r^,  the  minor  axis,  2b,  is  also  dependent  upon  the  direc- 
tion of  the  initial  velocity,  the  equation  of  condition  being 


Wl 


sin  m,,  (16) 


156  CALCULUS  OF    VARIATIONS. 

where  m^  is  the  angle  which  the  orbit  at  the  point  r^  makes 
with  r^ ;  and  this  element  of  direction  we  have  thus  far  entirely 
ignored.  If  now  we  assign  the  value  of  m^,  b  and  conse- 
quently e  will  be  given  by  (16),  and  ^  must  then  be  determmed 
by  making  the  ellipse  pass  through  the  two  fixed  pomts. 

When  a  has  its  least  value,  so  that  but  one  ellipse  can  be 
described,  the  chord  joining  the  two  fixed  points  is  evidently 
a  focal  chord ;  and  when  a  permits  two  ellipses  to  be  de- 
scribed, one  of  them  will  have  its  foci  upon  opposite  sides  of 
this  chord,  while  the  other  will  have  both  upon  the  same  side. 
This  distinction  is  important,  as  we  shall  subsequently  show 
by  Jacobi's  method  that  only  when  the  ellipse  is  of  the  latter 
species  does  it  render  the  action  a  minimum. 

123.  If,  with  a  fixed. value  of  r^  and  v^,  we  regard  m^  as 
variable,  and  for  each  value  of  m^  cause  the  second  fixed  point 
B  to  assume  the  corresponding  position,  which  would  render 
one  solution  only  possible,  the  point  B  will  itself  always  be 
found  upon  the  perimeter  of  an  ellipse.  For  there  being  but 
one  solution,  if  D  be  the  chord  joining  the  two  fixed  points, 
the  circles  described  as  above  will  just  touch  on  D,  and  we 
shall  have 

2a-r,-\-2a  —  r,—D,         or     Z>+r,  =  4^  — r„. 

But  D  and  r,  are  variable,  while  a  and  r^  are  constant.  There- 
fore, since  the  point  B  is  always  so  situated  that  the  sum  of  its 
distances  from  the  first  fixed  point  and  the  centre  of  force  is 
always  equal  to  a  constant,  it  is  on  an  ellipse  whose  foci  are 
at  these  two  points,  whose  major  axis  is  d,a  —  r^,  and  whose 

eccentricity  is  — ^- — ;  and  we  may  call  this  eUipse  the  limit- 

ing  ellipse. 

(24.  We  may,  in  closing,  advert  to  the  two  remaining 
cases  of  this  problem., 


PROBLEM  OF  LEAST  ACTLON.  157 

Suppose,  first,  that  we  make  v^  equal  to  r  — .      Then  it  is 

known  that  v'  will  equal  y  — ;  and  proceeding  precisely  as 

in  the  former  case,  or  better  by  making  C  zero  in  equation 

(14),  (since  that  equation  is  true  when  -  is  zero,)  we  shall 

obtain 

I  _  I  +^cos(^+^) 


r 


(17) 


the  equation  of  a  parabola,  in  which  b  is  one  half  the  latus 
rectum. 


Suppose,  secondly,  that  we  have  v^  =  y  ^^-\-:L,     Then 


=  i/'^+C 


/  2f  f 

we  know  that  v'  will  always  equal  r  —  +  -- ;  and  proceed- 

r         a 

ing  in  all  respects  as  before,  we  shall  obtain,  in  the  place  of 

equation  (14), 


^  =  ^  +  r  ^.+  <^cos(^+^),  (18) 

where   C  has  the  same  value  as  in  (14).     If  now  we  write 
b=^  —  a(\  —  e^),  (18)  will  readily  reduce  to 

£  ^  _  I  ±ccos{v-\-g) 

r  a(i-e')        '  ^^^^ 

But  we  shall,  in  the  course  of  the  investigation,  obtain  an 
equation  identical  in  form  with  (8),  except  that  W  will  equal 

y  — I — .     Hence,  that  c  or  Vb  may  be  real,  b  or  —  (i  —  e"") 
r       a  ^  ^ 

must  be  positive ;   and  therefore,  since  a  is  by  supposition 

positive,  it   readily  appears  that  i  —  ^'  is  negative ;  so  that 


158  CALCULUS  OF    VARIATIONS. 

since  e  in  this  case  is  greater  than  unity,  (19)  becomes  the 
equation  of  an  hyperbola,  having  its  attracting  focus  within 
the  curve.  This  is  as  it  should  be,  since  a  particle,  revolving 
in  an  orbit  according  to  the  Newtonian  law,  can  never  de- 
scribe an  hyperbolic  arc  having  the  attracting  focus  without 
the  curve. 

Problem  XXIII. 

125.  It  is  required  to  determine  the  form  of  the  plane  closed 
curve  of  given  length  which  will  envelop  a  maximiiin  area. 

Assume  the  pole  within  the  figure,  and  let  /  be  the  length 
of  the  given  perimeter.  Then,  because  the  curve  is  to  be 
closed,  we  have 

S/r^J^r'-'dv,  (i) 


which  is  to  remain  constant.    Now  m  being  the  enclosed  area, 

2 
we  have 


we  have,  by  the  principle  of  polar  areas,  dm  =  —  dv,  so  that 

r^r'dv  .        ,  , 


which  must  become  a  maximum. 

Now  the  reasoning  of  Bertrand,  in  Arts.  92  and  93,  is  evi- 
dently rendered  applicable  to  polar  co-ordinates  by  substi- 
tuting V,  r,  /,  etc.,  for  Xy  y,  y,  etc.  Whence  we  conclude  that 
Euler's  method  may  be  used  for  polar  co-ordinates  just  as  it 
has  been  hitherto  employed.  We  must,  then,  maximize  abso- 
lutely the  expression 


U=J      V-  +  ^V9^-\-r''\dv=l     Vdv.  (3) 


CLOSED   CURVE    OF  MAXIMUM  AREA,  159 

Here  F  is  a  function  of  r  and  r' ,  and 


ar' 


""-VTW^'  ^'' 


so  that  by  formula  (C),  Art.  56,  we  have 

2  ^         ^  VT+T"^  ' 


and 

r'-4-  ■    


'^+  -^J^T^'  =^'-  (5) 


Therefore 

=  7^-20,  (6) 


Whence 

^     4^V-    _  .  ^   .  4^V-  _  (^  -  2.y 
ir'-  2cY  {r'  -  2cy 

Hence 

dv  7^  —  2C 


dr       r  V4a'r'  -  {r'  -  2cf 


(7) 


(8) 


Now  squaring  r"  —  2c  under  the  radical  sign,  dividing  both 
numerator  and  denominator  by  r^^  and  then  placing  within 
the  radical  the  quantity  4^  —  4^,  (8)  may  be  written  thus : 


Write 


^  =  r  +  ^.  (,o) 


i6o 


CALCULUS  OF  VARIATIONS. 


Then  (9)  becomes 


dv  = 


dZ 


i/^a'  +  SC-Z' 
Therefore,  by  integration,  we  obtain 

Z 


and 


V  -\-g=i  sin" 


sin(t^+^) 


Z 


V4a'+Sc 


(II) 


(12) 


(13) 


Clearing  fractions  and  restoring  the  value  of  Z,  then  clearing 
fractions  again  and  transposing  the  first  member,  we  obtain 


r^  —  2r  Vd"  -\-  2c  sin  {v  ^  g)  -\-  2c  —  o, 


(H) 


which  is  one  form  of  the  polar  equation  of  the  circle  when 
the  pole  is  assumed  at  pleasure,  a  being  the  radius. 

(26.  Equation  (14)  is  the  form  in  which  the  result  is  left  by 
Prof.  Todhunter.  (See  his  History  of  Variations,  Art.  99.) 
To  interpret  this  result,  let  P  be  the  pole,  APB  a  diameter, 
and  denote  PA  by  C. 


Then  since  the  equation  of  the  circle,  when  the  origin  is  at-^, 
a  being  its.  radius,  is  y  =  2ax  —  4^^  if  we  remove  the  origin  to 
P,  it  will  become 

f=2a{x^C)-{x-^C)\  (15) 


CLOSED   CURVE   OF  MAXIMUM  AREA.  l6l 

Now,  in  passing  to  polar  co-ordinates,  let  r  =  PFbe  the  radius 
vector,  and  AB  the  initial  line.  Then  we  have  x  —  r  cos  v, 
and  y  —  r  sin  v.  Substituting  these  values  in  (15),  and  per- 
forming the  indicated  squaring,  we  easily  obtain  by  transpos- 
ing, observing  that  sin^  v  -\-  cos^  z^  =  i ,  . 

r"  —  2aC  —  C  Ar2r(a  —  C)  cos  v 


=  2aC—  C'-^2rVd'  -  2aC-\-C'  cos  ^.  (16) 

Now  put  2c  for  —  2aC-\-  O,  and  also  put  for  cos  v  the  sine  of 
its  complement,  v'.  Then  transposing  the  second  member  of 
(16),  and  putting  -j  for  v\  or  the  angle  DPY,  it  becomes 


r"  —  2r  V a" -\- 2c  sm  V -\-2c  =  o  \  (17) 

and  by  assuming  any  other  initial,  as  FG,  it  is  plain  that  the 
present  v  will  become  v  plus  some  constant,  say  g. 

■  127.  In  this  problem  the  terms  at  the  limits,  which  should 
be 

present  a  marked  peculiarity.  For,  since  the  curve  is  to  be 
closed,  we  must  consider  the  limits  of  integration,  viz.,  o  and 
27r,  to  be  fixed,  so  that  the  terms  become  merely  P^  Sr^  —  P^  Sr^. 
Moreover,  r^  and  r^  become  one  and  the  same  radius  vector, 
and  the  terms  at  the  limits  therefore  vanish  without  causing 
dr^,  Sr^,  Pj,  or  P^  to  vanish.  Hence  these  terms  furnish  no 
conditions  for  the  determination  of  the  arbitrary  constants 
which  enter  the  general  solution.  These  constants,  therefore, 
with  the  exception  of  a,  which  is  fixed  when  the  length  of  the 
curve  is  assigned,  must  remain  undetermined.  But  this  should 
not  be  otherwise.  For  we  see  from  the  last  article  that  g  is 
numerically  equal  to  the  angle  YPF,  while  c  depends  upon  the 
position  of  the  pole  with  relation  to  the  centre ;  and  we  can 


l62  CALCULUS  OF  VARIATIONS. 

evidently,  without  affecting  the  resuh,  assume  any  pole  and 
any  initial  line  we  please. 

If,  however,  we  had  required  that  a  curve  of  given  length 
should  pass  through  two  fixed  points,  and  should,  together 
with  the  radii  to  these  points,  include  a  maximum  area,  the 
three  constants  would  be  determined  from  the  assigned  length 
of  the  arc,  combined  with  the  two  equations  which  would 
hold  in  order  that  it  might  pass  through  the  two  given 
points. 

In  leaving  this  subject,  we  may  remark  that  whatever  has 
been  shown  concerning  the  general  treatment  of  the  limiting 
terms  in  problems  of  rectangular  co-ordinates  will  be  equally 
applicable  here.  Thus,  if  the  limiting  values  of  v  only  be 
assigned,  while  those  of  r,  r' ,  etc.,  are  subject  to  variation,  we 
must  equate  the  coefficients  of  ^r^,  ^r/,  dr^,  dr/,  etc.,  severally 
to  zero.  If  it  become  necessary  to  vary  the  limiting  values 
of  V  also,  we  change  v^  into  v^  -\-dv^,  and  v^  into  v^-\-dv^ ;  and 
if  the  required  curve  is  to  have  its  extremities  upon  two  other 
curves,  equations  (lo)  of  Art.  69,  or  the  more  simple  equations 
(2)  of  Art.  yOy  will  be  appHcable  w^hen  we  put  v  for  x,  r  for  j, 
r'  for  y ,  etc. 


Section  VIII. 


DISCRIMINATION  OF  MAXIMA    AND  MINIMA 
{JACOBUS    THEOREM). 

128.  We  have  already  seen  that,  in  discussing  the  maxi- 
mum or  minimum  state  of  any  definite  integral,  we  must 
equate  the  terms  of  the  first  order  in  its  variation  to  zero,  and 
then,  having  solved  the  differential  equation  obtained  thereby, 
this  solution  must,  if  it  do  not  reduce  the  terms  of  the  second 
order  to  zero  also,  render  them  positive  for  a  minimum  and 
negative  for  a  maximum.    We  have  also  seen  that  the  method 


JACOBTS  'theorem.  163 

of  transforming  these  terms,  so  as  to  render  their  sign  evident, 
has  been  far  from  uniform,  while  in  some  cases  we  have  been 
unable  to  investigate  the  sign  of  these  terms  at  all.  We  now 
proceed  to  explain  Jacobi's  Theorem,  which  gives  us  an  invari- 
able method  of  investigating  the  sign  of  these  terms  when  the 
limiting  values  of  jt,  7,  y' ,  etc.,  are  fixed.  But  as  the  general 
discussion  is  somewhat  abstruse,  Ave  shall  begin  with  the  most 
simple  case,  which  is  also  the  one  which  will  most  frequently 
present  itself  for  consideration. 

Case  i. 
Assume  the  equation 

u=iydx,  (I) 

where  V  is  any  function  of  x,  y  and  y'  only.  Then  to  the 
second  order,  inclusive,  we  have 

•^•^o     {dy     -^    ^    dy     ^    ) 

the  Hmiting  values  of  x  being  fixed.  Now  the  terms  of  the 
first  order,  when  transformed  in  the  usual  manner,  become 

P.  ^y.  -  P.  ^lo  +  r>  ^7  dx, 

where 

dy' '  dx       dy        dx  dy'' 

But  if  we  would  render  U  a  maximum  or  minimum,  the 
solution  of  our  problem  must  be  the  value  of  y  obtained  by 


164  *  CALCULUS  OF  VARIATIONS. 

completely  integrating  the  equation  M  =  O]  and  since  this  is 
an  equation  of  the  second  order,  this  value  of  y  will  certainly 
be  some  function  of  x  and  two  arbitrary  constants,  say 

y=f{x,c,,c,)  ^f.  (3) 

Of  course  other  constants  may  enter  F,  and  consequently  y, 
but  with  these  we  are  not  now  concerned.  Then,  since  the 
form  of  the  function  /  will  be  independent  of  the  conditions 
which  are  to  hold  at  the  limits,  we  must  next  so  determine  c^ 
and  c^  as  to  satisfy  these  conditions,  and  then  the  solution  be- 
comes complete  so  far  as  the  terms  of  the  first  order  are  con- 
cerned. 

129.  The  foregoing  considerations  will  prepare  us  for  the 
discussion  of  the  terms  of  the  second  order ;  but  before  enter- 
ing upon  the  explanation  of  Jacobi's  Theorem,  we  may  say  that 
its  object  in  the  present  case  is  to  put  the  terms  of  the  second 

order  under  the  form  -— j  multiplied  by  the  square  of  a  cer- 
dy 

tain  function,  and  also  to  determine  the  form  of  this  function. 

Now,  since  the  terms  of  the  first  order  must  vanish,  there 

remain  only  terms  of  the  second  and  higher  orders,  and  we 

may,  to  the  second. order,  write 

SU^-  fj\^^/  +  2bdy  6/  4-  cSy^)  dx,        '         (4) 

where  a,  b  and  c  have  the  values  shown  in  (2). 

Let  as  assume  that  (5y^,  Sy^  are  zero ;  then  we  shall  first 
show  that  d^^can  be  written  thus: 

where  A  and  A^  are  variable  functions,  the  suffix    i  having 


JACOBPS    THEOREM.  1 65 

no  reference  to  limits.     Observing  that  ^y  =  -3—,  we  have, 

ax 

by  parts, 

fcd/'dx  =  cSy'dy  -  f  Sy  -^  cS/.  dx,  (8) 

Also 

/■  b^y  dy'dx  =  bSy"^  —  j  dy  -—  bdy.dx 

=  ^^/  -  f^^^y  ^y'^i'^  -  f^  ^fd^'     (9) 

Hence 

2fbSy  Sy'dx  =  bdf  -  f'^  S/dx,  (lo) 

Therefore,    collecting   results,    arranging   and   factoring,    we 
have 

tf ;/  =  1  j  (M/).  -  (M/).  +  {c6y  d/\  ^  {cSy  6/1  } 

which,  when  we  make  Sy^  and  (^J/o  vanish,  gives  (5^^  in  the  re- 
quired form,  and 

dx 

130.  We  will  now  show,  in  the  second  place,  that  if  we 
vary  M,  we  may  also  write 

6M=AdyJy-^Afiy\  (12) 

dx  ^     ^ 

We  have 

.         dy        dxdy'  dx 


l66  CALCULUS  OF    VARIATIONS. 

Varying  the  first  term,  we  have  aSy  -j-  b^y' ;  and  varying  P,  we 

dP 
obtain  d^y  +  cSy' ,     Hence  the  variation  oi  —  -j-  (that  is,  the 

change  which  it  undergoes  from  changing  y  into  y  -\-  dy^  and 

y'  into  y  +  dy'^  is  —  -7-  {bSy  -\-  cdy'),  or,  by  differentiation, 


—  b^y' T-  ^y r-  cdy\ 

dx  dx 


Collecting  and  arranging,  we  have 

and  therefore  we  may,  if  Sy^  and  Sy^  vanish,  write 

6U=\lyM6ydx.  (14) 

131.  We  have  already  shown  that  if  the  terms  of  the  sec- 
ond order  in  df/ vanish,  we  shall  be  obliged  to  examine  those 
of  the  third;  and  as  these  will  not  usually  vanish,  but  will  be 
positive  or  negative  at  our  pleasure,  we  shall  be,  in  general, 
safe  in  assuming  that  in  this  case  we  have  neither  a  maximum 
nor  a  minimum  state  of  U.  But  it  is  evident  that  the  quantities 
A  and  A^  are  not  at  all  in  our  power,  so  that  unless  those 
quantities  vanish  of  themselves  the  terms  of  the  second  order 
can  only  be  made  to  disappear  by  the  assumption  of  suitable 
values  of  Sy  and  dy' . 

Now  let  u  be  such  a  quantity  as  will  satisfy  the  equation 

Au-\-—-Ay  =  o,-  (15) 

dx  • 


J  A  GOBI'S    THEOREM.  1 6/ 

where  u'  =  — -.   Then  it  is  clear  that  if  Sj/  throughout  the  defi- 
dx 

nite  integral  can  be  taken  equal  to  u,  or  to  ku,  where  k  is  any 

constant,  dUto  the  second  order  will  vanish.     Of  course  since 

dy  and  Sy'  must  be  infinitesimal,  k  must  be  also  infinitesimal, 

unless  ti  be  already  so. 

132.  We  will  next  determine  the  quantity  u,  as  we  shall 
then  be  better  able  to  see  how  it  may  be  employed.  We  have 
seen  that  the  value  of  y  obtained  by  the  complete  integration 
of  the  equation  M  =  o  will  be  of  the  form  y  —  f{x,  c^,  c^  =  /, 
and  that  this  value  of  y  vt^ill  satisfy  the  above  differential  equa- 
tion independently  of  the  value  of  c^  and  c^.  If,  therefore,  we 
make  any  changes  in  the  form  of  the  values  of  these  constants, 
the  resulting  changes  in  jj/  and  its  differential  coefficients,  while 
not  necessarily  zero,  will  not  prevent  these  quantities  from 
still  causing  M  to  vanish.  Now  suppose  we  change  c^  into 
c,  +  dc„  and  c^  into  c^  -\-  ^c^,  where  ^c,  and  ^c^  are  infinitesimal 
but  independent  constants.  Then  denoting  by  Sy  and  dy 
the  corresponding  changes  in  y  and  y\  we  shall  have 


and 


'>''£"■+'£"■  <■«) 


*>'  =  s(f*.+|*-)'  (■') 


Hence  these  values  of  Sy  and  ^y\  if  admissible  throughout 
the  range  of  integration,  will  render  the  corresponding  varia 
tion,  (S'M,  zero  throughout  those  limits,  and  will  also,  as  we 
see  from  (14),  render  S^U  zero.  But  we  shall  find  it  conveni- 
ent to  write 


1 68  CALCULUS  OF  VARIATLONS. 

where  k  =  dc^  and  /  =:  — ^ ;  and  as  dc^  and  ^c^  are  entirely  in- 
dependent, we  can  make  /  assume  any  real  and  constant  value 
we  please. 

We  conclude,  then,  from  (13)  and  (15),  that  the  general  value 
of  u,  if  not  infinitesimal,  is 

-=s;+'i-  <■») 

But  although  this  is  the  most  general  form  of  it,  it  is  evident 
that  we  need  not  vary  both  constants  in  /,  so  that  we  may 
have 

ku  =  -^'^c,  or     ku=^  -^  dc^.  (26) 

133.  Let  us  next  consider  whether  ku  can  be  an  admissible 
value  of  ^y  throughout  U\  because  if  it  can,  there  will  be  no 
need  of  any  further  transformation  of  the  terms  of  the  second 
order,  since  there  will  be  at  least  one  mode  of  varying  j  which 
will  cause  these  terms  to  vanish. 

We  observe,  first,  that  since  dy  and  Sy'  must  be  always  in- 
finitesimal, if  ku  be  an  admissible  variation  of  y  for  any  por- 
tion of  the  integral,  say  from  x^  to  x^,  u  and  u'  must  remain 
finite  throughout  these  limits. 

In  the  second  place,  if  ku  be  an  admissible  variation  of  y 
throughout  a  portion  only  of  the  required  curve,  say  from  x^ 
to  x^y  while  the  values  of  x^,y^,  x^,y,  are  fixed,  then  to  certainly 
make  the  terms  of  the  second  order  vanish  we  must  have  y^ 
and  JK3  also  fixed  ;  must  change  y  into  y  -\-  ku  throughout  the 
limits  x^  and  x^,  and  leave  the  rest  of  the  required  curve  un- 
varied. As  this  requires  that  u  shall  vanish,  both  when 
X  =  x^  and  when  x  =  x^,  and  as  dy  could  not  equal  ku  through- 
out any  limits  unless  u  vanish  at  both  those  limits,  we  con- 
clude generally  that  to  make  the  terms  of  the  second  order 


J  A  GOBI'S   THEOREM.  1 69 

disappear  by  the  use  of  kii  for  6y,  u  must  vanish  at  least  twice 
within  the  limits  of  integration. 

In  the  third  place,  if  either  of  the  quantities  -4-  or  -^,  which 

are  not  in  our  power,  vanish  twice  within  the  range  of  inte- 
gration, while  at  the  same  time  its  first  differential  coefficient 
with  respect  to  x  remains  always  finite,  we  can  make  the  terms 
of  the  second  order  disappear  by  putting  that  quantity  for  ?/, 
but  not  otherwise. 

Moreover,  that  we  may  employ  the  general  value  of  kti^ 

all  the  quantities  ^,  - — ^,  -^  and  — — ^  must  remain  finite 
dc^    dx  dc^    dc^  dx  dc^ 

throughout  the  limits  for  which  ku  is  employed,  and  we  must 

also  be  able  to  so  assume  ti  that  it  shall  vanish  at  least  twice 

as  we  pass  from  x^  to  x^. 

We  will  now  consider  under  what  circumstances  this  lat- 

di 

ter  condition  can  be  fulfilled.     Put  h  for  — ».      Then  we  see 

df_ 

dc^ 
from  (19)  that  we  can  cause  u  to  vanish  for  any  value  of  x  we 
please,  say  for  x  =  x^,  by  taking  /  =  —  K/,  and  this  is  all  that 
we  can  effect.  We  can,  moreover,  in  some  cases  assume  u 
so  that  it  shall  not  vanish  as  we  pass  from  x^  to  ;i-,,  while  in 
other  cases  we  cannot.  For  our  power  over  u  depends  en- 
tirely upon  our  assumption  of  /.  Now  suppose  we  find  that 
Ji,  which  is  not  in  our  power,  cannot  assume  all  possible  values 
from  negative  to  positive  infinity  as  we  pass  from  x^  to  x^. 
Then,  by  assuming  /  equal  to  one  of  these  values,  but  multi- 
plied by  —  I,  we  can  effect  that  u  shall  not  vanish  within  the 
Hmits  x^  and  x^.  But  if,  on  the  other  hand,  we  find  that  h 
ranges  through  all  real  values,  we  cannot  assume  /  so  that  u 
shall  not  vanish  at  least  once. 

To  apply  the  foregoing,  assume  /  so  that  u  shall  vanish 
when  X  —  x^.     Then  if  the  range  of  h  through  all  real  values 


I/O  CALCULUS  OF    VARLATIONS. 

be  complete,  il  will  evidently  vanish  again  at  or  before  the 
upper  limit,  according  as  h  may  complete  or  more  than  com- 
plete its  range,  and  we  can  make  the  terms  of  the  second 
order  vanish  by  the  use  of  kit.  But  if  the  range  of  Ji  be  only 
partial,  u  will  not  vanish  again  at  or  before  the  upper  limit, 
and  we  cannot  employ  ku  to  make  those  terms  disappear. 

(34,  It  is  evident  that  when  kit  cannot  be  employed  to 
make  the  terms  of  the  second  order  vanish,  some  further  trans- 
formation will  be  necessary  to  render  their  sign  apparent ;  and 
to  this  we  now  proceed. 

Let  u  involve  k — that  is,  be  ku — so  that  it  may  be  infinitesi- 
mal, and  resume  the  equations 

^u^\Jiy\^^y^^^^fiy'\^y'^^  (21) 

and 

Au-{-4-A,u'  =^0,  (22) 

ax 

Then  whatever  be  the  value  of  (^^',  we  may  certainly  make  it 
equal  to  ut,  and  (21)  will  then  become 

where  hii)'  =  -—  ut. 
ax 

We  wish  now  to  reduce  (23)  by  integrating  it  by  parts ; 

but  before  doing  so  we  must  show  that  because  (22)  is  true, 

the  expression 

u  -   AiitA^-—A,{ut)'  I  dx     or     Wdx  (24) 

can  always  be  integrated,  its  integral  taking  the  form  B^t' , 
where  B^  is  a  new  variable  function,  the  suffix  i  having  no  ref- 

reference  to  limits,  and  t'  =  -—. 

dx 


J  A  GOBI'S    THEOREM.  I7I 

135.  Multiply  (22)  by  tit,  and  subtract  the  product  from  the 
value  of  W  in  (24),  and  we  have 

W^u\±^Aiut)'-ut£-A,.'\.  (.5) 

Now 

u  —  A^itit)'  —  —-  uAltif)'  —  Alutyu.  (26) 

But  {lit)'  =  uf  +  ///.     Whence 


u4-Alut)'=.  ^u'A.t'^^  —tiA.tu'  -A.u'ut'  -A.u'H     (27) 
ax  ax  ax 


and 


Whence 


——itA^u't  =  iiA^t'  -\- 1  -—  uA^u' . 
ax  ax 


u  A.  ASuty  =  I  -^  uAy  -  Ay  \t+-f:  ^''^^^'-     (2^) 

ax  \  ax  J  a.X' 

Now  if  the  differentiation  indicated  in  the  first  member  of 
(28)  were  performed,  it  is  evident  that  the  only  term  in  which 
t  could  appear  undifferentiated  would  be 

ut-^Ay         or      \4-uAAi'  -A.iiAt, 
ax  {  dx  ) 

Hence  we  see  from  (25)  that  the  terms  in  li^  which  contain  t 
will  cancel,  and  we  shall  have 

V'^^ic'A,t'  =  ^Bjf, 
ax  dx 

where 

B,=  n'A,  (29) 

and 

/  Wdx  =/^_  B,t'.  d^  =  B/,  (30) 

the  constant  being  neglected. 


172  CALCULUS  OF    VARIATIONS. 

136.  By  the  use  of  (30),  (23)  may  now  be  integrated  by 
parts  thus : 


Wtdx 


2e/Xo 


(31) 


= \  \  ^tB/\  -  {tB,t\  I  -  \iyrdx. 


Now  examining  equations  (29),  (11),  (4)  and  (2),  we  see  that 

d'^V 
B,  =  u'A,  =  -u'c=-.^u'';  (32) 

and  since  we  put  Sf  equal  to  ut,  we  have 

,^usy-^^  (33) 

If  the  terms  without  the  integral  sign  in  (31)  do  not  vanish, 
they  must  be  added  to  those  already  in  (11).  But  the  suppo- 
sition that  6j/^  and  fy^  are  zero  will  certainly  reduce  these 
terms  to  zero  unless  21^  and  u^  vanish,  which  would,  as  we 
have  seen,  indicate  generally  that  there  is  neither  a  maximum 
nor  a  minimum.  Therefore,  finally  substituting  for  B^  and  /' 
their  values  from  (32)  and  (33),  we  have 


d/'  u' 


dx 


-2eAo  d/^ u^ ^-^'  (34) 

and  if  we  now  consider  ii  as  no  longer  involving  k,  we  must 
multiply  the  last  member  by  k^, 

137.  Let  us  now  consider  the  last  equation  more  particu- 
larly 


JACOBPS   THEOREM.  173 

First.  We  shall  assume  that  before  obtaining  this  equation 
it  had  been  ascertained  that  the  terms  of  the  second  order 
could  not  be  reduced  to  zero  by  any  use  of  ku  for  Sy ;  that  is, 
that  u  could  be  so  assumed  as  not  to  vanish  at  all,  since  other- 
wise the  last  transformation  would  be  needless.      / 

Second,  Now  suppose  the  second  factor  of  (24)  does  not 
vanish  permanently,  in  which  case  it  will  evidently  be  posi- 
tive ;  and  also  that  it  remains  finite  throughout  the  range  of 
integration.     Then  for  a  maximum  or  a  minimum  we  require 

d'^V 
only  that  -— ^  or  c  shall  remain  finite,  shall  not  vanish  perma- 
nently, and  shall  be  of  invariable  sign.  For  we  have  already 
seen  that  infinite  values  cause  the  method  of  development  em- 
ployed to  become  inapplicable,  and  even  in  the  case  of  a  single 
element  of  an  integral,  render  the  entire  result  doubtful.  More- 
over, if  c  can  change  its  sign,  we  can,  as  has  been  previously 
shown,  vary  y  for  such  values  of  x  as  will  render  c  negative, 
while  leaving  y  unvaried  for  all  other  values  of  x,  and  thus 
make  d^  negative ;  or  by  pursuing  a  similar  course  with  such 
values  of  x  as  render  c  positive,  we  can  make  dU  positive. 
But  if  c  remain  finite,  be  of  invariable  sign,  and  do  not  vanish 
permanently,  we  shall  have  a  maximum  or  a  minimum  accord- 
ing as  it  is  negative  or  positive. 

Third.  But  suppose  the  second  factor  of  (34)  does  vanish. 
Then  we  must  have 


Whence 


u'Sy  —  udy'  —  O.  (35) 

u'    J  Sy'    .  dti       dSy 

—  ^;r  =  -f-  dx,        or     —  =  — -^ . 
u  oy  li         dy 


Therefore  ISy  =  lu  -\-  g—  lu-^  Ik—  likii),  and  dy  —  ku,  where 
k  is  any  infinitesimal  constant.  But  by  supposition  the  prob- 
lem is  such  that  Sy  cannot  be  made  equal  to  ku  throughout 
the  range  of  integration,  and  therefore  the  second  factor  of 
(34)  will  not  vanish  permanently. 


174  CALCULUS   OF    VARLATLONS. 

Hence  we  see  that  if  the  terms  of  the  second  order  cannot 
be  reduced  to  zero  by  the  use  of  ku,  then  unless  c  vanish  they 
cannot  be  reduced  to  zero  by  any  admissible  mode  of  varying 
y,  and  this  supplies  what  was  before  wanting  in  the  complete 
investigation  of  the  subject.  To  render  the  second  factor  of 
(34)  infinite,  we  must,  if  dy  and  Sy'  be  infinitesimal,  have  either 
zi  =0  or  u'  —  00  .  But  the  first  condition  disappears,  since  we 
suppose  zi  to  be  taken  so  as  not  to  vanish  at  all,  and  the  second 

cannot  occur  unless  ^-  ^  or  -—  ^  become  infinite. 
ax  dc^       ax  dc^ 

It  will  be  seen  that  the  expression  uSy  —  Syti'  m  (34)  is 
the  determinant  of  u,  u' ,  Sy^  dy'\  so  that,  putting  D  for  their 
determinant,  we  may  write 


2  <^^o  u""  ' 


and  we  shall  see  hereafter  that  determinants  can  always  be 
employed  in  expressing  the  final  results  of  Jacobi's  transfor- 
mation. 

138.  Before  applying  this  theorem  to  any  example  the 
following  general  directions  may  be  useful. 

First.  Having  obtained  the  general  solution,  find  — ;-  or  c, 

dy 

which  must  not  vanish  permanently,  become  infinite,  nor 
change  its  sign.  For  in  the  first  case  the  terms  of  the  second 
order  would  reduce  to  zero;  in  the  second  the  investigation 
would  become  more  or  less  unsatisfactory  ;  while  in  the  third 
the  terms  of  the  second  order  can  be  made  to  assume  either 
sign,  thus  rendering  a  maximum  or  a  minimum  impossible. 

Second.  If  these  conditions  be  satisfactory,  find  the  quanti- 
ties -—  and  -5^,  neither  of  which  must  vanish  twice  within  the 
dc,  dc„ 


JACOBPS    THEOREM.  175 

range  of  integration,  otherwise  we  can  reduce  the  terms  of 
the  second  order  to  zero  by  employing  this  quantity  for  u. 

Third.  Moreover,  the  first  differential  coefficients  of  these 
quantities  with  respect  to  x  should  remain  finite  as  we  pass 
from  x^  to  x^,  otherwise  some  element  of  SU  mscy  become  infi- 
nite, thus  rendering  the  result  untrustworthy. 

Fourth.  If  all  these  conditions  still  indicate  a  maximum  or 
a  minimum,  consider  next  whether,  in  the  general  value  of  //, 

h  or  the  ratio  between  the  quantities  -4-  and  -4-  can  rang-e  over 

dc^  dc„_ 

all  real  values  as  we  pass  from  x^  to  x^.   For  if  it  can,  the  terms 

of  the  second  order  can  be  made  to  vanish  by  the  use  of  ku ; 

but  if  it  cannot,  those  terms  cannot  be  reduced  to  zero  by  any 

admissible  values  of  dy,  and  our  investigations  are  complete, 

assuring  us  of  a  maximum  or  a  minimum  according  as  c  is 

negative  or  positive. 


Problem  XXIV. 

139.  It  is  required  to  apply  Jacobi  s  Theorem  to  Prob.  I. 

Here  the  general  solution  is 

y  =  f{x,  c„  e,)  =  f=  c,x  +  ^,.  (i) 


Also, 

so  that 


d'V  I 


dy        |/(i_|.yy 


(2) 


and  this  last  expression  is  evidently  positive,  finite,  and  of  in- 
variable sign.     We  likewise  obtain  from  (i) 


176  CALCULUS  OF  VARIATIONS. 

df 

and  ir  =  i»  (4) 


and 


df  _ 

dc. 

I, 

d  df  _ 

dx  dc^ 

I, 

d  df  _ 

dx  dc„ 

0. 

(5) 


(6) 


Now  neither  of  the  first  two  quantities  can  vanish  twice,  nor 
do  their  first  differential  coefficients  become  infinite.  More- 
over, if  we  divide  the  first  of  these  quantities  by  the  second, 
we  find  h  =  x,  which  will  not  range  through  all  real  values. 
Hence  u  can  be  so  assumed  as  not  to  vanish  at  all.  For  we 
have  zi  ^^  X  -\- 1]  and  bv  assuming  /  to  be  negative  and  numeri- 
cally greater  than  x^,  the  truth  of  the  assertion  becomes  evi- 
dent. Jacobi's  Theorem,  therefore,  indicates  a  minimum  in 
this  case. 

Problem  XXV. 

140,  //  is  required  to  apply  the  theorem  of  J ac  obi  to  the  case 
of  the  brachistochrone  in  Prob.  II.,  Case  i. 

Here,  from  equation  (11),  Art.   17,  the  general  solution, 
which  is  a  cycloid,  is  seen  to  be  of  the  form 


y  —f{x,  c„  c^=f—  c,  versin-  -  --  V2c^x  —  x''  +  c^,    (i) 
where  c,  is  the  radius  of  the  generating  circle.     We  also  have 


Vx 
so  that 

^^=  _  '  (2) 


J  A  GOBI'S   THEOREM.  1/7 

This  last  expression  is  of  invariable  sign  and  positive,  but  be- 
comes infinite  at  the  cusp,  where  both  x  and  y'  are  zero.  The 
investigation  will  therefore  be  subject  to  any  doubt  which 
may  arise  from  this  fact,     (See  closing  remark  of  Art.  21.) 

Disregarding  this  objection,  we  have  from  (i),  by  differentia 
ating  carefully  with  respect  to  c^  and  c^  successively,  while 
treating  ^  as  a  constant, 

-f-  =  versin ,  (3) 


df_^ 
dc„ 


(4) 


'2 


Now  we  shall  take  x^  to  be  somewhat  less  than  2c^.  For,  as  we 
have  seen,  y'  becomes  infinite  at  the  vertex,  and  we  wish  as 
far  as  possible  to  avoid  infinite  quantities,  since  Jacobi's 
method  does  not  enable  us  to  overcome  the  obstacle  which 
these  quantities  present  to  a  satisfactory  solution.  With  this 
limitation  neither  of  the  above  quantities  will  vanish  twice 
within  the  range  of  integration.  We  also  have,  by  differen- 
tiating in  the  usual  way, 

d  df  _  -\ri  ^^, 

dx  dc^       {2c ^  —  x)^ 

d  df  _ 


dx  dc„ 


o,  (6) 


and  these  quantities  remain  finite  throughout  the  present  lim- 
its.    Moreover,  if  we  divide  -f-  by  -/-,  the  quotient  /i  will  be 

dc^         dc^ 

the  second  member  of  (3),  and  this  cannot  range  over  all  real 
values,  so  that  ii  can  be  so  taken  as  not  to  vanish  at  all  is  we  pass 
from  x^  to  x^.  We  conclude,  therefore,  that,  setting  aside  the 
objection  previously  mentioned,  Jacobi's  Theorem  indicates  a 
minimum  in  the  present  case. 


1/8  CALCULUS  OF  VARLATIONS. 

Problem  XX VI. 

141.  It  is  required  to  apply  the  theorem  ofjacobi  to  Prob.  XXI I. 

From  what  has  been  previously  said  regarding  the  treat- 
ment of  polar  co-ordinates  by  the  calculus  of  variations,  it  will 
appear  that  all  the  reasoning  by  which  Jacobi's  transformations 
were  effected  will  apply  also  to  them  when  we  change  x  into  v^ 
y  into  r,  and  y'  into  r' .  We  shall  consider  only  the  case  in 
which  we  have  an  ellipse,  our  object  being  to  verify  the  clos- 
ing remark  of  Art.  122.  We  shall,  with  slight  deviations,  fol- 
low Prof.  Todhunter.  (See  his  Researches ;  or  Adams  Essay, 
Art.  183.) 

Here,  as  we  see  from  equation  (5),  Art.  121, 


V=V---  Vr'  + 


Whence 


i/l-L 

dW       ^   r       a 


dr"        ^{r'^r'y 

which  cannot  change  its  sign,  and  is  always  finite  and  positive. 
Now  the  general  solution  in  equation  (15),  Art.  121,  may  be 
written 

v/  X        /-  ail  —  e^)  ,  . 

.=/(.,..,.,)=/=-^^-^,  (I) 

where  e  may  take  the  place  of  c^^  and  g  that  of  c^. 

It  appears  that  (i)  contains  also  another  constant,  a.  But 
this  constant  was  introduced  when  we  assigned  the  initial 
velocity,  and  is  not  therefore  a  constant  of  integration.  Now 
we  have  already  stated  that /might  involve,  besides  the  inde- 
pendent variable  and  c^  and  c,^,  any  number  of  other  constants; 


JACOBVS    THEOREM.  179 

those  only  which  enter  by  integration  being  considered  by 
Jacobi's  method. 

We  must,  then,  pursue  the  usual  course,  and  find  the  dif- 
ferential coefficient  of  /,  that  is,  of  r  with  respect  to  e  and  g. 
We  have,  from  (i), 

^SLufl  =  I  4-  ^  cos  {v  +  g),  (2) 

r 

Now  differentiating  with  respect  to  e,  we  obtain 

r  r"        de  _  ^     '  ^  ^       e   {         r  ) 

the  last  member  being  found  from  (2).     Solving  (3),  we  finally 
obtain 


and 


Also, 


dr  _r^  —  ar{i  -\-  e^) 
de  ae{i  —  e") 

d  dr  _  [2r-a(i+e')y 
dv  de  ae{\  — '/) 


(4) 
(5) 


dr  _dr  _    , 

dv  dg  ^^^ 

Now  neither  the  first  member  of  (5)  nor  (7)  can  become  infi- 
nite, so  that  we  may  employ  Jacobi's  Theorem  with  confidence. 
But  before  resorting  to  the  most  general  method,  let  us 
determine  whether  the  first  member  of  (4)  or  (6)  can  vanish 

dr 
twice.     Now  to  make  — -  vanish,  we  must  have 

de 

r  =  «(i  +  .').  (8) 


l80  CALCULUS  OF  VARIATIONS. 

But  this  is  the  value  of  the  radius  vector  drawn  to  the  ex- 
treraity  of  the  remote  latus  rectum.  For  the  distance  between 
the  foci  being  2ae,  and  the  semi-latus  rectum  being  a{\  —  e^), 
we  have 

^^  ^  4^V"  +  ^^  (i  -  ej  ^^a'^i^  ej. 


dr 
Also  r' ,  and  consequently  -— ,  vanishes  at  each  vertex  of  the 

dg 
ellipse,  so  that  we  conclude  at  once  that  there  will  be  no  mini- 
mum if  the  arc  extend  from  vertex  to  vertex,  or  be  cut  off  by 
the  remote  latus  rectum. 

Now,  in  applying  the  general  method,  we  are  only  con- 
cerned in  knowing  the  range  of  h,  or  the  ratio  of   ;    to  -^. 

dc^       dc„ 


But  h  evidently  varies  as 


But 


Whence 


• 3—^-  or    , (9 

r  r 


I  __  I  4"  ^  cos  {v  +  g) 


r 

a{i-e^) 

r' 

esm(v+g-) 

r' 

a{i-e^)    ' 

and  therefore  the  last  member  of  (9)  may  be  written 


i-^(i+.o 

^sin(^4-<^) 


(10) 


J  A  GOBI'S   THEOREM.  l8: 

Now  this  expression  varies  only  as 


(II) 


sin  {v-^g)  rsm{v-\-,g) 

Next  let  us  write 

r  =  2a  —  R,  (12) 

rsin(^  +  ^)  =  i^sin  w.  (13) 

Then  R  will  be  the  radius  vector  drawn  from  the  other  focus, 
and  w  will  become  the  angle  which  R  makes  with  the  major 
axis.     Then,  by  substitution,  (11)  will  become 

p   .    =  -. ]  -^-^ — -  -  il^ecotw,       (14) 

Rsmw  sm  ^  (        K  ) 

the  last  member  being  obtained  by  substituting  for  R  its  value 

— ^^ —,  whence  h  varies  as  cot  w. 

I  -\-  e  cos  w 

Now,  in  general,  any  function  will  have  a  complete  range 
from  negative  to  positive  infinity  when  we  can  cause  it  to 
start  with  a  given  value,  change  sign  by  passing  through  zero 
or  infinity,  and  return  to  its  initial  value.  But  cot  w  be- 
comes infinite  at  the  two  vertices  only,  vanishes  only  when 
r  is  the  semi-latus  rectum,  and  changes  sign  at  these  four 
points,  and  at  these  only. 

Now  let  i?o  and  R^  be  the  radii  drawn  to  the  two  fixed 
points.  Then,  to  make  cot  w^  and  cot  w^  equal,  r^  and  r,  must 
form  a  continuous  line ;  that  is,  a  focal  chord.  Should  the  arc 
extend  from  one  vertex  to  the  other,  cot  w^  and  cot  w,  will 
not  be  equal,  but  will  be  infinite  and  of  contrary  sign,  having 
passed  through  zero.  But  in  all  other  cases  cot  w^  and  cot  zv^ 
are  equal,  after  having  changed  sign  by  passing  through  in- 
finity. ^ 


l82 


CALCULUS   OF  VARIATIONS. 


Here,  therefore,  there  is  no  minimum,  and  if  the  arc  be  still 
greater  the  same  remark  will  hold,  unless  we  were  required 
to  vary  the  entire  arc.  For  since  we  can  make  it  vanish  at 
each  end  of  the  focal  chord,  we  can  take  dy  =  ku  through  that 
portion  of  the  arc,  and  leave  the  remainder  unvaried,  thus 
making  the  terms  of  the  second  order  in  (5^  t^  vanish.  But  if 
the  arc  be  less  than  that  subtended  by  a  focal  chord  passing 
through  the  present,  which  is  the  remote  focus — that  is,  both 
foci  lie  upon  the  same  side  of  the  line  joining  the  two  fixed 
points — then  the  range  of  cot  w  will  be  only  partial,  and  there 
will  be  a  minimum. 

142.  We  may  give  a  general  geometrical  illustration  of 
Jacobi's  method.  Let  A  and  B  be  two  fixed  points,  joined  by 
a  curve  which  satisfies  the  differential  equation  J/  =  o,  and 
let  CED  be  another  curve  derived  from  the  first  by  such  vari- 
ations of  y  and  y'  as  will  result  from  varying  the  constants  of 
integration,  and  consequently  still  satisfying  the  same  differ- 
ential equation. 


Then  there  will,  if  -r-w  permit,  be  a  maximum  or  a  mini- 
dy 

mum  when  CED  cannot  twice  meet  AB  unproduced.  But  if 
it  can  meet  it  twice,  we  may  regard  AFEGB  as  the  new  de- 
rived curve,  which  would  make  the  terms  of  the  second  order 
vanish. 

But  since  we  can  make  u  vanish  once  at  pleasure,  we  may 
suppose  the  derived  curve  to  touch  the  other  at  A — that  is, 
we  can  make  C  and  A  coincide — and  then  we  shall  have  a 


J  A  GOBI'S    THEOREM.  1 83 

maximum  or  a  minimum  so  long  as  the  other  point  of  meet- 
ing, G,  is  not  reached. 

Moreover,  we  compare  AB  with  such  derived  curves  onl}^ 
as  satisfy  the  equation  M  —  o,  ahhough  their  number  may  be 
infinite.  For  we  have  seen  that  when  ku  cannot  be  used  to 
make  the  terms  of  the  ^second  order  disappear,  they  will  not 

72  T" 

vanish  at  all  if  -—^r  do  not  vanish.     Hence  no  other  class  of 
ay 

curves  could  render  SU  \.o  the  second  order  zero. 

(4-3.  Now  it  is  evident  that,  in  order  to  employ  the  pre- 
ceding theorem,  we  must  be  able  to  find  the  functions  -—  and 

— ;  that  is,  to  determine  the  change  which  y  would  undergo 

when  in  the  general  solution  we  give  infinitesimal  increments 
to  c^  and  c^.  We  therefore  naturally  first  seek  to  obtain  the 
complete  integral  of  the  differential  equation  J/  =  o,  and  to 
exhibit  it  under  the  form  of  j/  =zf{x^  c^,  c^. 

But  it  frequently  happens  that  even  when  we  are  unable 
to  obtain  the  general  solution  in  the  explicit  form  just  given, 

we  can  still  determine  the  functions  -^  and  -^.    Still  this  is  not 

dc^  dc^ 

strange,  since  we  can  often  obtain  the  differential  of  an  un- 
known quantity  ;  that  is,  a  differential  whose  integral  is  unob- 
tainable. When  these  functions  can  be  found,  Jacobi's  method 
can  be  applied  to  the  inves,tigation  of  the  terms  of  the  sec- 
ond order,  whether  the  equation  M  =  o  can  be  completely 
integrated  or  not ;  and  we  now  proceed  to  show  how  they 
may  be  determined  in  the  case  of  a  very  important  class  of 
problems. 

The  following  method  is  due  to  Prof.  Todhunter  (see  his 
Researches,  Arts.  26,  282),  and  we  shall  see  that  by  it  he  has 
been  able  to  obtain  some  results  not  previously  known,  and 
to  correct  some  which  had  been  erroneously  given. 


1 84  CALCULUS  OF  VARIATIONS. 


Problem  XXVII. 


144-.  It  is  required  to  discuss  in  full  the  conditions  which  will 
maximize  or  minimize  the  expression 


U=  r^y^  V  dx  =  r^  Vdx, 


Xa 

where  v  is  any  function  of  y'  and  constants. 
Here  F  is  a  function  of  y  and  y'  only,  and 

dy 
Hence,  by  formula  (C),  Art.  56,  we  have 

whence 

f'{v-y'v')^c,,  (I) 

which  is  as  far  as  the  integration  can  be  carried,  so  long  as  n 
and  V  are  entirely  undetermined.  But  we  may  suppose  a 
curve  to  be  drawn  satisfying  (i),  and  that  its  equation  is 
y  z=i  fix,  c^,  c^  =/.  Then,  although  we  cannot  determine  the 
form  of/,  we  can  ascertain  what  would  be  the  corresponding 
variation  oi  y  if  c^  and  c^  were  increased  by  Sc^  and  Sc^,  and  can 
then  investigate  the  terms  of  the  second  order. 

145.  From  (i)  we  have 


Also, 


;'=f-^=/(/>0=/.  (2) 

v  —  y  v 


y       0   dy' 


JACOBVS    THEOREM.  1 85 

Whence,  by  supposing  the  integration  performed,  we  may 
write 

X  =  F{y\c,)  +c,  =  F-\-  c,.  (3) 

Now,  although  /  and  F  mayjcontain  other  constants  besides  c^, 
these  will  not  be  affected  by  any  variation  of  c^  or  c^,  leaving 
only  y'  and  c^  as  variables.  Moreover,  x  will  undergo  no 
change  when  c^  and  c^  vary,  and  these  constants  themselves 
are  entirely  independent  of  each  other.  We  have  then,  from 
(2)  and  (3), 

dc,       dc,~^dydc,  ^^^ 


and 


Whence 


_dF      dFdy 
^~  dc,'^~dy'~d^^'  ^^^ 


d_ld_l  ^_dF^ 

dy'  dc^  dc^  ^  ' 

Differentiating  (2)  and  (3)  with  respect  to  x,  we  obtain 

and 
Whence 

Hence,  and  then  multiplying  by  (8)  and  comparing  with  (6), 

df  dy'  _  y'  dy'  _  y'dF  dy'  _       y'dF 


dy  dc^       /'  dc,  dy'  dc,  dc. 


(10) 


1 86  CALCULUS  OF   VARIATIONS. 

Therefore 

dy  _df      y'dF 
dc^       dc^        dc^  ' 

Again,  from  (2)  and  (3),  we  have 

dy  _df  dy' 
dc^       dy'  dc^ 
and 


00 


(12) 


dFdy'   ,  ,     , 


dy  

Whence,  by  (9), 


Comparing  this   equation   with    (8),  we   obtain   -^  —  —  y' 

dc„ 


!;=-/•  ('4) 

We  must  next  determine  the  form  of  f~-J  and  f-^J,  which 

are  only  partial  differentials  with  respect  to  c^^  this  fact  being 
indicated  by  writing  them  in  brackets. 
From  (2)  we  have 

where  m  —  -.     Hence 
n 

\dcj        {y — yv'f  ^     ^ 

But  from  (i)  we  have 

^  m         r  m 
V  -^       J  ymn  y  ' 

and  therefore,  restoring  n,  w^e  have 


dcj       ncl 


(17) 


JACOB/' S    THEOREM.  1 87 

Now  although  we  cannot,  while  v  is  unknown,  determine  F, 
still  it  is  evident,  from  its  mode  of  derivation  from  /,  that  if 
Cj^  enter  the  latter  as  a  factor,  it  must  also  enter  the  for- 
mer unchanged.  F  must  therefore  be  of  the  form  c;"^w,  where 
w  is  some  function  not  involving  c^  or  c^,  but  merely  y\  and 
perhaps  constants,  not  of  integration.  Hence,  from  (3),  we 
have  ' 

x  =  c,^w-{-c^  (18) 

and 

X  —  c^ 

w  = -. 


Now 


Therefore,  finally, 


dF^ 

=  wmcJ^-^  —  -,  (10) 


dy  y  —  y'{x  —  c^ 

dc,  nc^ 


(20) 


(46.  Now  if  the  value  of  y  found  by  the  solution  of  (i) 

can  render   U  a  maximum  or  a  minimum,  the  terms  of  the 

second  order  in  dC/can  be  put  under  the  form  given  in  equa- 

d'^V        y'^d'^v 
tion  (34),  Art.  136.     Then,  supposing  -— ^  or  —r-jr  ^^  ^"^''  ^^ 

be  of  invariable  sign  and  finite,  it  will  only  be  necessary  that 
u  shall  be  incapable  of  vanishing  twice ;  which  will  in  general, 
as  we  have  seen,  follow  if  it  can  be  so  taken  as  not  to  vanish 
at  all.  Now  equations  (14)  and  (20)  give  us  the  general  value 
of  2/,  thus : 

dc^       dc^  dc^        dc^         dc^ 


where  L  =  ncd. 


1 88  CALCULUS  OF  VARLATLONS. 

Now  by  differentiating  the  last  equation  with  respect  to  x, 
it  will  at  once  appear  that  u'  will  not  become  infinite  so  long 
as  y"  is  finite — that  is,  so  long  as  there  occur  no  cusps.  Were 
this  not  so,  we  could  not  feel  entire  confidence  in  the  follow- 
ing investigations. 

But  in  order  to  make  u  vanish  without  supposing  either 

of  the  quantities  -^  or  -f-  to  vanish,  we  must  have 

;tr  —  ^  =  r,  —  Z.  (22) 

Now  if  y  be  the  ordinate  of  the  curve,  we  know  that  the  first 
member  of  (22)  will  represent  the  abscissa  of  the  point  in  which 
the  tangent  to  the  curve  at  y  will  meet  the  axis  of  x,  and  we 
will  denote  this  abscissa  by  X.  But  since  Z  is  a  constant 
entirely  in  our  power,  we  can  give  to  the  second  member  of 
(22)  any  value  we  please.  If,  therefore,  there  be  any  real 
value  which  X  cannot  assume,  we  can,  by  making  the  second 
member  take  that  value,  render  equation  (22)  impossible,  and 
thus  cause  that  u  shall  not  vanish  at  all. 

But  suppose  either  of  the  quantities  ~-  or.  -5-  to  vanish 

twice.     Then  equating  the  first  to  zero,  we  obtain  x  —  -^  =  c^. 

Whence,  if  it  vanish  twice,  there  must  be  two  tangents  which 
meet  on  the  axis  of  x  at  the  point  whose  abscissa  is  c^.  That 
the  second  quantity  may  vanish  twice,  y  must  also  vanish 
twice. 

(47.  We  may  now  complete  the  discussion  of  Prob.  VIII., 
as  promised  in  the  closing  remark  of  Art.  63, 

Here  n  is  unity,  and /of  that  article  is  identical  with  t/. 
Suppose,  as  before,  that  y  is  positive,  but  that  the  curve,  in- 
stead of  being  concave,  is  always  convex  to  the  axis  of  x. 
Then  X  cannot  always  range  over  all  real  values.     For  sup- 


J  A  GOBI'S    THEOREM,  ■  1 89 

pose  the  line  AE  to  slide  as  a  tangent  along  the  curve  from  A 
to  B.  Then  if  we  assume  DE  as  the  axis  of  ;r,  this  line  cannot 
meet  x  between  D  and  E,  and  the  range  of  X  is  not  therefore 
complete.  But  if  CE  be  the  axis  of  ;r,  X  will  assume  all  real 
values,  its  range  being  just  complete ;  while  if  GH  be  taken  as 
the  axis  of  ;r,  then  X,  having  passed  through  infinity,  will 
complete  its  range  before  B  is  reached,  and  will  then  repeat 
the  values  of  ;r  from  G  to  N.  If  we  consider  such  an  arc  as 
BK,  the  range  of  X  will  evidently  be  restricted,  and  the  tan- 
gents at  B  and  K  will  intersect  above  K — that  is,  above  x — 
since  the  ordinate  of  K  must  be  positive. 


^^~\ 


Hence  when  y'^  is  positive,  if  the  tangents  at  the  extremi- 
ties of  the  arc  intersect  above  the  axis  of  x,  we  shall  have  a 
maximum  or  a  minimum  according  as  v'^  is  negative  or  posi- 
tive, because  j/  is  positive,  and  we  have  seen  (Art.  63)  that 
wheny^  is  of  invariable  sign, /^',  which  is  here  u'^  will  be  also. 
But  if  the  extreme  tangents  intersect  on  or  below  the  axis  of 
X,  there  can  be  neither  a  maximum  nor  a  minimum. 


Problem  XXVIII. 

14-8.  B  is  required  by  means  of  the  preceding  method  to  apply 
Jacobis  Theorem  to  Prob.  VI I. 

Here  the  general  equation  to  be  considered  is 
U=    r^yVT+Y'dx^    r^yvdx. 


190  CALCULUS  OF  VARIATLONS. 

Whence  v"  =  -_===^^y  a  positive  quantity  ;  and  as  the  gen- 
eral solution  is  a  catenary,  having  the  directrix  as  the  axis  of 
X,  y  is  always  positive.  Therefore  we  infer  that  the  solution 
will  render  U  a  minimum  Avhen  the  extreme  tangents  intersect 
above  the  axis  of  x,  but  not  otherwise. 

Suppose,  then,  the  same  condittons  and  notation  as  in  Art. 
61,  which  will  of  course  hold  even  should  j^  and  j^  become 
equal.     Now  the  equations  of  the  extreme  tangents  are 

J/  —  I;  =  J//  (jr  —  c)  and    J  —  /^  =  jFo'  (-^  +  ^). 

From  these  equations  we  obtain 

W    y  —  k  —  y,'c' 

and  solving  for  y,  and  giving  it  a  suffix,  because  it  will  then  be 
the  ordinate  of  the  point  of  intersection  only,  we  have 

'- — jr^' — •  ^"^ 

Now  put 

2c  _2c 

L  —  e'^  —  e    «.  (2) 

I    - 
Then  multiply  equation  (4),  Art.  61,  by  y  ^",  equation  (5)  by 

I     _  ^ 

je~a^  subtracting  the  second  product  from  the  first,  and  then, 

observing  that  the  first  member  of  the  resulting  equation  be- 
comes identical  with  the  second  member  of  equation  (i)  of  the 
same  article,  we  have,  as  the  equation  of  the  catenary, 


y=      }c^lb(^  —  ke    ~''\+c    '"'Ike''  —  be    «  j  ^ 


(3) 


JACOBPS    THEOREM.  I9I 

Now  differentiating  (3)  with  respect  to  x  only,  and  then  sub- 

c  ex 

stituting  successively  in  the  result  e^  and  e   ^  for  ^,  we  have 

.      Mb-2k 

(4) 

(5) 


where 
Therefore 

But 


yr  = 

La      ' 

v'  — 

2b -Mk 

Jo    — 

La      ' 

M=^ 

2c                2c 

(6) 
f/-W={M-2){d  +  k)±.  (;) 

Z^  =  J/^  -  4  =  {M+  2)  {M-  2).  (8) 


Whence  M—  2  must  be  positive ;  and  as  L  cannot  become 
negative,  (7)  must  also  be  positive.  Multiplying  (4)  by  k,  (5) 
by  b,  and  subtracting,  and  then  multiplying  (4)  by  (5),  we  have 
the  equations 

ky,  -  by^  = ^;    ^  (9) 

and 

^  2M{b^-\-k^)-bk{^  +  M^) 

_  2M{U^  +  k^)  -  bk{A  +  2M^  -  M^) 

~  zv  ^    ^ 

Multiplying  (10)  by  2^,  adding  to  (9),  reducing  to  a  common 
denominator,  and  factoring,  we  have 

2cy:y,'  +  ky,'  -  by,'  = 
^\{b'-\-^  -  Mbk)  (2Mc  -La)  +  {M'  -  ^)  cbkX  .       (11) 


192  CALCULUS  OF  VARIATLONS. 

But  performing  the  multiplication  indicated  in  the  second 
member  of  equation  (6),  Art.  61,  it  may  be  written 

^^Mbk-{b'-^k').  (12) 

Hence,  and  recollecting  that  M'^  —  4=  L\  the  second  member 
of  (11)  will  become 

La       ,,    .    2cdk  .     ^ 

-^Mc  +  —  .  (13) 

But  equation  (8),  Art.  61,  may  be  written 

and  hence,  since  L  is  always  positive,  the  sign  of  (13),  and  con- 
sequently that  of  j/2,  the  ordinate  of  the  point  in  which  the 
extreme  tangents  intersect,  will  be  like  that  olF\ 

Now  it  was  shown  that  when  but  one  catenary  can  be 
drawn,  F^  is  zero ;  but  that  when  two  catenaries  can  be  drawn, 
F^  will  be  positive  for  the  upper  and  negative  for  the  lower. 
Hence  the  extreme  tangents  to  the  upper  catenary  will  inter- 
sect above  the  axis  of  x,  thus  giving  us  a  minimum ;  while 
those  to  the  lower  will  intersect  below  that  axis,  and  will  not 
give  a  minimum.  When  but  one  catenary  can  be  drawn,  the 
extreme  .tangents  will  intersect  on  the  directrix,  and  we  shall 
not  have  a  minimum.  Indeed,  we  may  here  suppose  that  the 
two  catenaries  coincide ;  and  for  a  demonstration  of  the  fact 
that  the  extreme  tangents  w^ould  in  this  case  intersect  on  the 
directrix,  see  Todhunter's  Researches,  Art.  72. 


JACOBVS    THEOREM.  1 93 

Problem  XXIX. 

149.  //  is  required  to  apply  the  general  method  of  Art.  146  to 
Case  2,  Prob.  II. 

Here  n= and  v=  Vi  +J^ so  that 

2 

^"V       „    „  I 

dy  |/_^(i  j^yy 

which  is  always  positive  and  finite ;  thus  indicating  a  mini- 
mum, so  far  as  it  is  concerned.  Now  as  the  general  solution 
in  this  case  is  a  cycloid,  having  the  horizontal  as  the  axis  of  x, 
we  know  that  X  cannot  assume  all  possible  values,  since  no 
tangent  can  meet  the  axis  of  x  within  the  cycloid.  Hence, 
without  determining  jj/  as  a  function  of  x,  or  even  obtaining 
the  value  of  u,  we  are  able  easily  to  apply  the  method  of 
Jacobi,  and  to  see  that  we  have  a  minimum. 

This  result  is,  however,  subject  to  any  doubt  which  may 
arise  from  the  fact  that  y  is  infinite  at  either  cusp,  but  is  alto- 
gether trustworthy  so  long  as  the  portion  of  the  curve  which 
we  are  considering  does  not  contain  any  cusp,  as  will  be  the 
case  if  the  particle  is  to  start  with  an  initial  velocity. 


Problem  XXX. 

(50.  It  is  required  to  apply  the  theorem  of  Jacobi  to  Prob. 
XVL 

Here,  as  will  be  seen  from  equation  (8),  Art.  98,  the  gen- 
eral solution  is  a  sphere,  having  its  centre  upon  the  axis  of  x ; 
and,  recollecting  that  y  must  not  become  negative,  that  equa- 
tion may  be  written 


y=VAd'-{x-c:f.  (I) 


194  CALCULUS  OF  VARIATIONS. 

Now  it  must  be  observed  that  a  is  not  a  constant  of  integra- 
tion, but  was  introduced  in  accordance  with  Euler's  method 
for  treating  problems  of  relative  maxima  and  minima,  so  that 
it  cannot  be  varied  in  applying  Jacobi's  Theorem  ;  and  func- 
tions involving  it,  together  with  x,  j/,  c^  and  c^,  will  merely  be 
mentioned  as  functions  of  the  latter  quantities. 

It  appears,  then,  that  y  has  in  this  case  been  obtained  mere- 
ly as  a  function  of  x  and  c^,  it  having  been  necessary  in  equa- 
tion (3),  Art.  98,  to  make  the  first  constant  of  integration  zero 
before  we  could  effect  the  second  integration.  Since,  there- 
fore, the  constant  c^  has  disappeared  from  the  value  of  j/,  we 

dy  dy 

shall  not  be  able  readily  to  obtain  the  functions  -f-  and  -7-  re- 

dc^  dc^ 

quired  in  the  application  of  Jacobi's  Theorem. 


151.  Since  we  have  seen  (Art.  99)  that  the  sign  of  2a  must 
be  negative,  we  have  from  equation  (i),  Art.  98, 


F  =  /-  2ay^/\^y'\ 

Therefore 

(rV_ 2ay 

dy^~     V(i+/7' 

which,  being  negative,  indicates,  so  far  as  it  is  concerned,  that 
the  volume  is  a  maximum. 

Now  observing  the  sign  of  2a,  equation  (3),  Art.  98,  may  be 
v/ritten 

-^^==/  +  ^.  (2) 

Vi-\-y'' 

But  from  (2)  we  see  that  /  can  be  expressed  as  an  explicit 
function  oi  y  and  c^\  and  we  have  always 

dx   or   -^=f{y,c,)dy,  (3) 

y 


JACOBVS   THEOREM.  195 

Whence,  supposing  the  integration  to  have  been  performed, 
we  have 

'*^=/UO  +  ^.=/+^.-  (4) 

Therefore  —  must  in  any  case  equal  — .    Taking  the  total  dif- 

dy  y 

ferential  of  (4)  with  respect  to  c„  recollecting  that  any  change 
in  c^  will  affect  y  but  not  x,  we  have 


Hence 


dc^       dy  dc^       dc^       y'  dc^ 

^  =  -y^.  (6) 

dc^  dc^ 


Now  in  hke  manner,  recollecting  that  c^  does  not  occur  ex- 
plicitly in  /,  we  have 


and  therefore 


dy  dc^  y  dc^ 


%.'-"■  <'■' 


We  must  now  determine  the  value  of  -—-,  observing  that  it  is 

dc^ 

only  the  partial  differential  coefficient  of /with  respect  to  c,. 

If  /  could  be  found  as  an  explicit  function  of  y  and  c^,  this 

could  be  done  directly ;  but  as  /  cannot  be  so  found,  we  must 

adopt  an  indirect  method.     Now  the  supposition  \\\'dXy  is  to 

become  constant,  and  c^  variable,  will  make  dy  constant,  but  y' 

still  variable,  because  it  is  capable  of  being  expressed  as  an 

explicit  function  of  y  and  <:,,  although  -^  will  be  no  longer 

total,  but   merely  partial,  and   can   be  at  once  found.     But 


196  CALCULUS  OF  VARIATIONS. 

f—  /— 7-;  and  if  in  this  expression  we  vary  ^,,  regarding jj/  as 


y 

constant,  and  indicate  partial  differentials  by  brackets,  we 
shall  have 


But  in  this  case  we  must  have  dy=z     -^     dc^\  and  as  6c^  must 
be  constant,  we  have 


rdf- 


Now  from  (2),  by  partial  differentiation,  we  obtain 

2ayy'       [dy'^_ 


4/(1 +y^) 


Hence 


152.  When  the  general  solution  is  a  sphere,  this  integral 
can  be  obtained.  For  if  in  (2)  we  put  r  for  2a,  make  c^  zero, 
and  divide  by  y,  it  will  become  the  differential  equation  of 
the  circle,  whose  centre  is  on  the  axis  of  x ;  and  we  shall  have 

and     dy^  '^''^^ 


Hence  (10)  becomes 


[l]=-r/^^>- 


JACOBVS   THEOREM.  197 

dzv 

Now  put  y  =  tan  w  and  dy'  = r — .     Then 

cos  w 

rdfl  _       I    f        dzv  _       ^    r  cos'  w  +  sin''  w  , 

\_dc^j  r^  cos  w  sin'  w  r^      cos  w  sin"  2x/ 

I   j    f*  cos  w  ,  '         r   dw    \  . 

= 1    /   -^-^dzv-\-       \.  (12) 

r  ( *^    sin  zi;  ^    cos  w  )  ^ 

Now  by  integrating  this  expression,  we  shall  obtain 

r^l  =  -ii  I  I  ,i+sin7c>]  ^  _i_^^ 

U/^  J  ^   I       sin  2£/    '   2     I  —  sin  zt/  j  r     * 

Hence,  finally,  by  equation  (6),  we  have 


^  =  --?^'  Z. 
dc^       r 


(14) 


It  will  then  at  once  appear,  by  comparing  (7)  and  (14),  that  the 
range  which  we  are  in  this  case  to  examine  will  be  entirely 

dependent  upon  that  of  Z.     Now  when  zt/  is  -,  Zis  —  00  ;  and 

when  w  is  zero,  ^  is  +  °^  *»  so  that  Z  ranges  twice  from  —  00  to 
-|-  00  as  we  pass  from  x^  to  x^.  We  would  therefore  naturally 
infer,  from  the  employment  of  Jacobi's  method,  that  the  sphere 
is  not  the  solid  of  revolution  whose  volume  for  a  given  sur- 
face is  a  maximum ;  an  inference  which  we  know  to  be  erro- 
neous. 

153.  Although  for  convenience  we  have  hitherto  tacitly 
assumed  that,  even  when  the  terms  of  the  second  order  are  to 
be  considered,  we  may  by  Euler's  method  convert  any  prob- 
lem of  relative  maxima  or  minima  into  one  of  absolute  max- 
ima or  minima,  we  have  not  yet  established  the  correctness  of 
this  assumption  ;  while  we  see  from  the  last  article  that  it  can- 


19^  CALCULUS  OF  VARIATIONS. 

not  be  universally  true.  In  order  to  discuss  the  subject  in  a 
general  manner,  let  us  resume  the  conditions  and  notation  at 
the  beginning  of  Art.  92.     Then,  as  there,  we  shall  have 

/      6vdx=:  Vdydx        and       /     dv' dx  =    /      Vdydx. 

Moreover,  since  the  limiting  values  of  Sj/,  dy,  etc.,  are  to  van- 
ish, the  terms  of  the  second  order  will  become 

-T' 6  Vdydx  and     -T'd  Vdydx, 

This  we  have  already  seen  to  be  the  case  when  the  func- 
tion contains  no  differential  coefficient  higher  than  y' ,  and  we 
shall  subsequently  see  that  it  is  true  generally. 

It  must  likewise  be  observed  that  now,  besides  being  infini- 
tesimal, the  variations  of  j,  y' ,  etc.,  are  restricted  to  such  sys- 

tems  of  values  as  will  render  /     v'dx  constant ;  and  although 

U  Xq 

we  cannot  express  explicitly  the  nature  of  this  restriction,  and 
although  the  systems  of  values  which  it  permits  for  Sy,  6y\ 
etc.,  may  still  be  infinite  in  number,  it  cannot  be  disregarded 
in  the  discussion  of  the  problem. 

We  shall  denote  this  restriction  by  writing  the  variations 
affected  in  brackets ;  then,  to  the  second  order,  we  have 

Xyv^ dx ^/;'  v\sy^ dx + \iy v]  m dx=k^i  (o 

and 

£ySv'^ dx  =£'  V'idy]  dx  +  \S^ySV'^  \Py\  dx^m+  n.  (2) 

Now  since  /  'vdx  is  to  be  a  relative  maximum  or  minimum, 
k  -\- 1  must  certainly  be  a  small  negative  or  positive  quantity 


JACOBPS   THEOREM.  I99 

of   the   second   order ;    and   since   /    v'dx  is  to  undergo  no 

change  when  y,  y\  etc.,  are  varied,  m  -\-  71  must  vanish,  at  least 
so  far  as  any  quantity  of  the  second  order  is  concerned. 

154-.  Thus  far  there  can  be  no  doubt;  but  what  follows 
may  perhaps  be  subject  to  some  criticism,  as  the  author  has 
not  seen  it  in  any  other  v/ork,  although  he  will  not  assert  that 
no  similar  discussion  occurs. 

Now  the  equation  m  =  —  n  must  be  true  to  the  second 
order,  so  that  it  appears  that  in  need  not  vanish  absolutely, 
but  must  become  less  than  any  quantity  of  the  first  order ;  and 
we  are  therefore  led  to  infer  that  k  also  will  not  vanish,  but 
become  a  quantity  of  the  second  order.  That  this  supposition 
is  not  inadmissible  in  problems  of  relative  maxima  and  min- 
ima, we  have  already  seen  in  the  beginning  of  Art.  94.  But 
these  suppositions  regarding  k  and  m  will  not  invahdate  the 

V 
reasoning  of  Art.  92,  by  which  it  was  shown  that /or--  must 

be  a  constant ;  because  /  could  not  differ  from  a  constant  by 
any  finite  quantity. 

Now  assume  the  equation 

C"dx+b£yjdx^u,  (3) 

where  b  is  any  constant  whatever.    Then,  since  J     v'  dx  is  to 

undergo  no  change  when  we  vary  jj/,y,  etc.,  the  variation  of 
u  to  any  order,  as  the  second,  will  to  that  order  equal  merely 
the  variation  of  its  first  term.     Hence  we  may  write 

£\sv-\dx=£;[v-^bv'\\sy\dx 

+  \.C  {  \.^n+b[SV'^  \  [Sy-\dx  =  k^-^bm  +  l-\-bn.      (4) 


200  CALCULUS  OF    VARIATIONS. 

Now  so  long  as  b  remains  undetermined,  k  -f-  bm  may  be  a 
quantity  of  the  second  order ;  but  when,  as  explained  in  Art. 

92,  we  put  b  —  a  —  —  y-,  we  effect  that  k  -(-  am  shall  certainly 

vanish,  since  those  terms  are  then  equivalent  to 

Therefore  we  have 

^ + '^^  =  \S.l^  [\pn+'^\p  n  ]  i^j'i  dx,       (5) 

as  the  exact  expression  to  the  second  order  of  the  change 
which  /  vdx  will  experience  when  y,  y',  etc.,  are  varied  ac- 
cording to  the  conditions  of  the  problem ;  and  this  is  the  only ' 
mode  of  rendering  the  expression  exact,  since  it  is  not  only 
sufficient,  but  also  necessary,  that  b  should  become  a  in  order 
to  make  the  terms  of  the  first  order  entirely  vanish. 

Now  according  to   Euler's  method,  let  U  be  what  u  be- 
comes when  b  —  a.     Then  to  the  second  order  we  have 

\s£yod^  =  \SU^  =  ij(""  i  [cJF]  +«  [tfF']  I  \_Sy-\  dx.       (6) 

Whence  it  appears  that  we  can  and  must  employ  Euler's 
method  to  obtain  the  terms  of  the  second  order  in  an  explicit 
form.  But  it  will  be  observed  that  the  restriction  still  adheres 
to  the  variations  in  (6),  and  no  method  of  further  determining 
its  effect  upon  the  general  form  of  (^C/has  yet  been  devised; 
still,  if,  as  is  usually  the  case,  the  general  solution  can  render 
the  second  member  of  (6)  invariably  negative  or  positive  for 
unrestricted  values  of  Sy,  Sy\  etc.,  this  restriction  can,  of 
course,  exercise  no  influence  upon  the  problem,  and  we  shall 
be  certain  of  a  maximum  or  a  minimum.     But  if,  on  the  other 


JA GOBI'S    THEOREM.  201 

hand,  by  employing  the  most  general  values  of  Sy,  S/,  etc.,  it 
should  be  found  possible  to  cause  the  second  member  of  (6) 
to  assume  either  sign  or  to  vanish,  we  may  conclude  justly 
that  t/is  not  an  absolute  maximum  or  minimum.  But  this 
will  not  warrant  us  in  asserting  that   [/,  and  consequently 


t/a;o 


vdx,  may  not  be  a  relative  maximum  or  minimum ;  that  is, 

a  maximum  or  minimum  for  all  such  values  of  Sj/,  6j/\  etc.,  as 

will  render  /   '  v'dx  constant ;  and  having  no  means  of  taking 

propor  account  of  this  restriction  upon  the  variations,  we  may, 
at  least  theoretically,  be  unable  to  determine  whether  U  is  or 
is  not  a  relative  maximum  or  minimum. 

155.  Thus  we  see,  first,  that  Euler  s  method  must  be  em- 
ployed in  developing  the  terms  of  the  second  order  in  this 
•class  of  problems ;  and  if  by  it  we  seem  to  have  a  maximum 
or  a  minimum,  we  may  accept  the  decision  as  final.  But  if,  on 
the  contrary,  we  appear  to  have  neither  a  maximum  nor  a 
minimum,  we  cannot  always  conclude  that  such  is  really  the 
case,  the  discrimination  being  correct  as  regards  an  absolute, 
but  perhaps  not  as  regards  a  relative  maximum  or  minimum 
state  of  ^. 

This  latter  result  is  mentioned  by  Prof.  Todhunter  (see  his 
Researches,  Art.  283) ;  and  evident  as  it  is,  when  the  former 
is  admitted,  it  appears  not  to  have  been  noticed  by  any  pre- 
vious writer.  The  former  result,  however,  is  assumed  by  him 
without  proof.  Prof.  Jellett  has  given  no  discussion  of  the 
ierms  of  the  second  order  in  questions  of  this  character. 

156.  We  can  now  understand  why  the  theorem  of  Jacobi 
is  not  as  satisfactory  for  problems  of  relative  as  for  those  of 
absolute  maxima  and  minima.  For  example,  in  the  preceding 
problem  the  condition  that  the  surface  is  to  remain  constant 
will  prevent  us  from  making  Sy  invariably  positive  or  nega- 
tive ;  and  as  it  must  change  sign,  it  will  certainly  vanish  at 


202  CALCULUS  OF   VARIATIONS. 

least  once  between  the  limits  x^  and  x^,  say  at  the  point  whose 
co-ordinates  are  x^  and  y^.  But  even  if  we  can  so  select  x^  that 
u  can  vanish  both  when  x  =  x^  and  x  =  x^,  as  we  certainly  can 
by  considering  a  hemisphere,  it  does  not  follow  that  we  can 
ma*ke  the  terms  of  the  second  order  throughout  the  integral 
vanish  by  the  use  of  kit.  For  when  we  assume  dy  =  ku 
throughout  the  first  hemisphere,  we  may  be  obliged  to  make 
some  change  in  the  form  of  the  other  also ;  that  is,  ku  may 
not  be  an  admissible  value  of  ^y  unless  the  first  hemisphere 
be  permitted  to  increase  or  diminish  its  surface. 

Nevertheless,  when  Jacobi's  method  seems  to  indicate  a 
maximum  or  a  minimum,  that  indication  may  be  regarded  as 
trustworthy. 

157.  We  may,  in  passing,  notice  two  particular  and  ex- 
ceptional cases  which  may  arise  in  the  general  application  of 
this  theorem.  These  cases  appear  to  have  befn  first  noticed 
by  Spitzer.      (See  Todhunter's   History  of  Variations,  Arts. 

173,  174.)     Suppose,  first,  that  -—^  =  0  throughout  the  inte- 

dy 

gral.  Now  if  V  involve  y'  at  all,  it  can,  to  render  this  equa- 
tion true,  contain  only  its  first  power.  Therefore  the  general 
form  of  V  must  be 

V=f{x,y)^y'F{x,y)=f^-y'F.  (i) 

We  shall  write  total  differentials  in  brackets.     Then 

U^  r'  Vdx, 

the  limiting  values  of  x  and  y  being  fixed  ;  and  therefore  to  the 
first  order  we  have 


JA GOBI'S   THEOREM.  203 

Therefore,  as  usual,  we  obtain 
But 


Vdx\~  dx~^^  dy' 


dy 
so  that  (3)  becomes 

df      dF 

i-^=^-  (4) 

Now  (4)  involving  only  x,y,  and  possibly  constants,  which  are 
not  of  integration,  we  can,  by  solving  for  j/,  obtain  it  as  a  func- 
tion of  X  without  constants  of  integration.  Hence,  in  appli- 
cations to  geometry,  it  will  be  impossible  to  satisfy  the  gen- 
eral solution  unless  the  given  points  happen  to  be  situated 
upon  the  curve  which  is  determined  by  (4). 
The  second  case  is  that  in  which  we  have 


d'V  .     d'V 

o,         and 


dy"         '  d/        Ldxdydy'} 


d    dW-]  _ 


o. 


As  this  case  is  more  difficult  than  the  former,  and  is  rather 
curious  than  important,  we  shall  merely  give  its  interpreta- 
tion without  proof. 

First,  /  being  some  function  of  x  and  y,  f  and  f"  being 
functions  of  x  only,  and  the  differentials  not  enclosed  in  brack- 
ets being  partial,  it  is  shown  that  Fmust  have  the  general 
form 

Whence 


204  CALCULUS  OF    VARIATIONS. 

Therefore 

Hence  if  ^is  to  be  a  maximum  or  a  minimum,/'  must  vanish 
for  all  values  of  x^  and  U  must  be  of  the  general  form 

which,  since  the  last  integral  is  constant  and  might  be  written 
F{x),  is  not  a  general  problem  of  variations.  Thus  in  this  case 
the  maximum  or  minimum  value  of  f/must  be  sought,  if  at 
all,  by  the  differential  calculus;  and  if  the  limiting  values  of 
X  and  y  be  fixed,  U  can  have  no  maximum  or  minimum  state. 

In  both  these  cases  F  involves  the  first  power  only  of  y, 
and  they  are  therefore  examples  of  Exception  2,  Art.  51. 

158.  We  may  now,  before  considering  the  next  case,  pre- 
sent the  following  general  view  of  the  treatment  of  the  terms 
of  the  second  order  according  to  Jacobi. 

Assume  the  equation  U=J  Vdx,  where  V  is  any  func- 
tion of  X,  J,  y  .  .  .  .  y'^\  and  regard  the  limiting  values  of 
X,  y,  y  .  ,  .  .  y**- 1)  as  fixed.  Then,  as  before,  the  solution  must 
be  obtained  from  the  differential  equation  M  —  o,  which  will 
in  general  be  of  the  order  27t.  Hence  its  complete  integral 
will  involve  2n  arbitrary  constants,  and  may  be  written 

and  this  solution  is  rendered  complete  when  the  constants  are 
so  determined  as  to  satisfy  the  conditions  at  the  limits. 

159.  Next  the  terms  of  the  second  order  must  equal 

T     r^f 

-J^    SMdydx. 


JACOBFS    THEOREM.  205 

For  we  have  always 

But  if  we  vary  these  coefficients,  leaving  dy,  Sy' ,  etc.,  unvaried, 
we  shall  obtain  the  well-known  form  for  the  terms  of  the  sec- 
ond order  in  dV\  namely, 

df    ^  ^  dydy'  ^  ^  ^  dy"    ^    ^  ^  dy^^''-  -^ 

Therefore  it  appears  that  the  terms  of  the  second  order  in  d^ 
must  in  any  case  equal  half  of  what  would  result  from  varying 
those  of  the  first  order,  supposing  Sy,  <5>',  etc.,  to  undergo  no 
change.  They  should  not,  however,  be  considered  as  really 
arising  in  this  manner,  as  y,  y' ,  etc.,  receive  no  second  incre- 
ment.    But  when  the  limiting  values  of  y,  y\  etc.,  are  fixed, 

the  terms  of  the  first  order  in  (^^  become  /  MSydx^  so  that 
those  of  the  second  order  must  equal -/     SMSydx. 

160.  It  is  evident  that  the  reasoning  of  Art.  132  would  be 
equally  applicable  whatever  might  be  the  order  of  the  difler- 
ential  equation  M=  o,  and  we  shall  therefore  assume  at  once 

that  SMsLud  J^  '  dMdydx  will  vanish  if  for  Sy  we  substitute 


the  variations  of  c^,  c^,  etc.,  being,  as  before,  entirely  indepen- 

dent.     Then  Sy',  Sy\  etc.,  will  become  — -  or  ?/,  -— -  or  n",  etc., 

dx  dx 

the  differentials  being  total  with  respect  to  x. 


206  CALCULUS  OF  VARIATIONS. 

It  will  next  be  shown  that  SM  can  be  made  to  assume  the 
form 

dM  ^  Ady  A^  ^  Afiy'  -\.  eiQ ^^A^d^^\ 


After  this  the  terms  of  the  second  order  can  be  integrated 

I    r^^  d'  V 
by   parts,  until  they  finally  take  the  form  SU=  -J      t-t^c 

multiplied  by  the  square  of  a  certain  function,  analogous  to 
that  previously  found. 

As  the  proof  of  the  last  two  pomts  is  necessarily  difficult, 
the  general  reader  may,  without  serious  loss,  omit  the  re- 
mainder of  this  theorem,  or  may  at  least  assume  the  truth  of 
the  two  following  lemmas,  whose  use  will  be  at  once  evident. 


Lemma  I. 
(61.  SM  C2ixv  always  be  put  under  the  form 

SM=A8y^^^  A,dy  +  etc +  ^"L ^^  ^Z"*^- 

We  shall,  for  convenience,  abandon  our  former  notation, 
and,  adopting  that  of  Prof.  Jellett,  write 


Whence 

SM—dN  —  d         .   ^_ ^_ 

dx  dx 


dP  d'^P 


Si\ —1  -\-  etc ±  — - — - .  (2) 

dx    ^  dx'^  ^  ^ 


JACOBVS   THEOREM.  20J 

Jm  p 

For  take  any  term  as  6  -— ^^  =  dti'^),  where  t  =  P„^.      Now 

if  in  Art.  9,  we  put  /  for  j/,  ^''  for/,  etc.,  recollecting  that  /',  t\ 
etc.,  are  the  total  differential  coefficients  of  t  with  respect  to 
X,  we  shall,  by  reasoning  precisely  like  that  there  employed, 

find  that  dt^'^'^  =  -^-;  so  that  it  is  evident  that  (2)  has  been 
correctly  transformed.     But  • 

,,.„  =  f|..^  +  ^.y  +  etc +p,<^/''>, 

and 

p    _   dV 

^  ~  ~d/^)' 

Therefore 
Hence 


m 


^;ir"^  ^;i'"^  (  dy^'^'^dy  dy^'^^dy 

.        dW 


'  dy^'^^dy^'^^ 
Now  consider  some  individual  term  of  this  series,  as 

rim  //2  77  ^m 

^j^m  ^j,{m)  ^y{l)    -y  dx"^        -^  ^^^ 

d^V 
where  /  is  not  screater  than  7n,  and  k  =    ,     ■■  ,  ,,,.     Now  if  / 
^  dy^^^dy^^) 

equal  ;;/,  this  term  is  already  under  the  required  form ;  but  if 

/  be  less  than  m,  there  will  certainly  arise  from  the  develop- 

d^Pi  d^ 

ment  of  — i-  a  term  of  the  form  -— ,  >^  (5>("^),  the  sisfn  of  these 
dx^  dx'' 


208  CALCULUS  OF  VARIATIONS. 

terms  being  like  or  unlike,  according  2iS  m  —  I  is  even  or  odd. 
That  is,  if  SM  be  fully  written  out,  it  will  be  found  that  with 
the  exception  of  those  terms  which  are  already  under  the  re- 
quired form,  all  the  others  may  be  arranged  in  pairs,  the  type 
of  which  is  the  pair 

But  by  a  theorem  of  the  differential  calculus,  any  pair  of 
the  form  (4)  can  be  arranged  in  a  series  of  the  form 

(See  Note  to  Lemma  I.) 

Whence  it  appears  that  all  the  terms  in  dj/can  be  ar- 
ranged as  stated  at  the  beginning  of  this  lemma. 

Lemma  IL 

162.    If  A,  A^,  etc.,  be  functions  of  x,  implicit  or  expHcit, 
and  u  any  quantity  which  will  satisfy  the  equation 

7  72 

A2^+4-  ^y + 4t  ^y + etc. = o  -,  (o 

ax  ax 

then  if  we  write 

U^  u  {  Aut-^--^Aiuty^j-Alut)"^^tz.  I  ,         (2) 

Udx  will  always  be  integrable  whatever  be  the  value  of  /,  the 
integral  taking  the  form 

J  Udx  =  B/  +  ^  B/'+  etc.,  (3) 

where  B^,  B^,  etc.,  are  functions  derived  from  A,  A^,  etc. 


JACOBrS    THEOREM.  209 

As  the  proof  of  this  theorem  belongs  entirely  to  the  in- 
tegral calculus,  we  follow  the  plan  of  Prof.  Jellett,  and  append 
it  in  a  note  (see  Note  to  Lemma  II.). 

Case  2. 

163.  Next  let   U  —  J     Vdx,  where   V  is  any  function  of 

X,  y,  y'  and  y'\  the  limiting  values  of  x,  y  and  y'  being  fixed. 
Then,  proceeding  in  the  usual  manner,  the  general  solu- 
tion must  be  found   from  the   differential   equation  M  =  o, 
where 

dV       d  dV       d'  dV 
~  dy       dx  dy'  "^  dx'  dy"'  ^^^ 

The  complete  integral  of  (i)  will  give  7  in  the  form 

y=f{^x,  c,,  c^,  c„  c,)  =/,  (2) 

in  which  the  four  constants  must  be  so  determined  as  to  satisfy 
the  given  values  of  y^,  y^,  y/,  yj. 

But  when  these  limiting  values  are  fixed,  we  need  not  ex- 
press the  terms  of  the  second  order  in  the  usual  way,  which 
expression  would  be  difficult  to  transform ;  but  we  may  write 
at  once 

SU=lXyMSydx.  .  (3) 

We  have  now  an  invariable  method  of  transforming  SU,  since 
we  can  always,  according  to  Lemma  I.,  put  dM  under  the 
form 

6M=Ady-\-^A,dyJr-^Afy\ 

and  we  shall  now  proceed  to  apply  this  lemma  in  order  to 
determine  the  functions^,  A^  and  A^. 


2IO  CALCULUS  OF  VARLATIONS. 

[64-.  For  brevity  of  notation,  let  ayy^  ayy>,  ay>y',  ayy,  ay^ 
and  Uy^y  denote  the  second  differentials  of  V  with  regard  re- 
spectively to  y,  y  and  y ,  y ,  y  and  y" ,  y'  and  y ,  and  y" .  Then, 
referring  to  the  value  of  i^  in  (i),  and  writing  its  variation 
in  full,  recollecting  that  the  variation  of  the  differential  of  any 
quantity  equals  the  differential  of  the  variation  of  that  quan- 
tity, we  have 

SM  =  Uyy  dy  +  ayy,  (5>'+  ayy  dy"  -  —  {ay,y  dy  +  ay,y,  (5>'+  a^y  6y") 

=  ayydy  -  —  ayy  dy  +  —  ayy  6/  +  [-  —  ayy, Sy  +  ayy.  dy' 
+  (£.  ayy  dy  +  ayy  dy')  +  g,  a^y  dy'  -  ^  a^y  dy') 


"^  -"^ dx\   ^yy^-" )  '  dx'  "^"^  ■"  '  \dx 


+  (£^,,,^  +  ,..y)  +  g...y-^.3c^r),        (4) 


where 


Now  the  first  three  terms  of  (4)  are  already  in  the  required 
form,  so  that,  setting  these  aside,  we  will  consider  the  first 
couple.  Here  /  — o,  n=  i,  and  there  can  be  but  one  term 
resulting  from  this  pair.  Therefore,  by  equation  (13),  Note 
to  Lemma  I.,  the  couple  becomes 

^c^^y^'\         or     c^dy,         or    -£.:;,^K^y^ 

or     ^ak,.dy,         or     -j- {- ayy).dy,  (6) 

dx  ax 


JACOB!' S   THEOREM.  211 

because  a  is  always  unity.  Now  consider  the  next  couple. 
Here  /  =  o,  n=  2,  and  the  number  of  terms  which  will  re- 
sult is  two.     Hence,  by  (13),  the  pair  becomes 

We  also  have  by  equation  (14)  of  the  same  note,  since  a  is  al- 
ways one,  and  b  is  in  this  case  two, 


and 


'''y+T/''y'='^'y+ii.^'^^y' 


^       2ayy6/.  (7) 


dx^  dx 

In  the  last  pair  we  have  /=  i,  ?/  =  i,  and  it  becomes 

dx\dx   ^  I       dx\  dx     -^  J  ^  ^ 

Collecting  results  from  the  last  members  of  (4),  (6),  (7)  and 
(8),  and  arranging,  we  have 


dx 

Afiy'  +  i^A,,/, 

where 

dayy,                 d'ayy. 

or        ttyy  —    a'yy,    +  a"  yy* , 

^.-^^.V  +  ^  +  2^..-> 

or          —  ay'y>  +  a '  y'y  -\-2ayy; 

• 

A^  ^ayy. 

(9) 

212  CALCULUS  OF  VARLATIONS, 

165.  We  may  now  write 


and  we  know  that  if  ii  be  an  admissible  value  of  Sy,  u  having 
the  form  given  in  Art.  i6o,  (^^can  be  rendered  zero,  and  we 
infer,  as  in  the  first  case,  that  there  will  be  neither  a  maximum 
nor  a  minimum.  But  since  the  limiting  values  of  y  and  y'  are 
to  remain  fixed,  we  must,  in  order  that  ^y  may  equal  u,  or  ku, 
be  able  to  so  determine  .the  constants  dc^,  dc^,  etc.,  that  both  u 
and  ic'  shall  vanish  twice  simultaneously  at  or  within  the  lim- 
its of  integration.  In  the  former  case  we  may  change  y  into 
y-\-  ku  throughout  the  limits,  while  in  the  latter  we  make  this 
change  merely  for  the  limits  at  which  u  and  u'  vanish,  leaving 
y  unvaried  throughout  the  remainder  of  the  integral.  Also, 
since  the  variations  of  y,  y'  and  y"  must  be  infinitesimal,  to 
make  ^y  equal  ku,  we  must  have  u,  u'  and  u"  finite  throughout 
the  limits  for  which  they  are  employed. 

166.  But  suppose  that  the  terms  of  the  second  order  can- 
not be  made  to  vanish  by  the  use  of  u.  Then  if,  as  before,  we 
put  ut  for  6y,  (lo)  will  become 

zrri/       Itdx,  (I  I) 

in  which  we  know,  from  Lemma  II.,  that  Idx  is  immediately 
integrable,  giving 

f  Idx  =  B/  +  4-  ^"^"-  (i2) 

'-^  dx      ' 

167.'  Let*  us  next  determine  the  functions  B^  and  B^. 
From  (lo)  and  (i  i)  we  have 


JACOBPS   THEOREM.  21 3 

^«  +  -^^,«'+^^,^<"  =  o  (13) 

ax  ax' 

and 

I=^Aut  +  ^-  ASut)'^  ^1  Alut)"  \  u.  (14) 

Whence,  multiplying  (13)  by  ut,  and  subtracting  from  (14),  we 
have 

1=  u4-A,{uty  +  7c-f^A,{uty  -  ut-^Ay  -  ut^-Ay 

dx  dx  dx  dx 

=  u  [AXutyy  +  u  {A,{utyY-ut{AX}'  -  ut{Ayy.   (15) 

Now  we  know  from  Note  to  Lemma  II.  that  all  the  terms  in  / 
which  contain  t  undifferentiated  must  eventually  cancel,  so 
that  we  may  neglect  the  last  two  terms  in  (15),  and  may  also 
reject  all  others  in  /  as  they  arise.     We  have  then 

u\Ai2ityY^\2iAStityY  -  u'Aluty  and      {tity  =  ut'-\-ti't. 

Whence 

u\Aluty \ '  =  {iiA.ut'y  +  {uA,  7i'ty  -  21' a,  uf  -  u'Aji't. 

Now  the  second  and  third  terms  of  this  equation  can  be  united 
into  one  by  Note  to  Lemma  I.,  because  here  /  =  o  and  n  ■=^  \. 
But  as  this  term  would  certainly  contain  /  undifferentiated, 
we  need  not  perform  the  operation,  but  may  reject  them  to- 
gether with  the  last,  retaining  only 

(«M,o'.  (16) 

Again,  we  have 

u\Aiuty'\"-=\uAi2ity'\"-  2  \u'Aiuty\'-^ti"Aiuty' 

and 

{Uty  ^  2a"-\-22c't'+lc"t\ 


214  CALCULUS  OF  VARIATIONS, 

SO  that 

u\ASut)"\"^  {uA,ut")"^2{uA,u'ty-\-(uAyt)"-  2{u'A,ut")' 

-  ^{tt'A^ u't')'-2{ii'A^ u"ty+  2i"A,ut"-\- 2u"A, u't'^ u"A, u"t.  (17) 

Now  set  aside  the  first  and  fifth  terms,  which  are  already  in- 
tegrable ;  reject  the  last,  and  also  the  couple  6  and  8,  because 
they  could  be  united  into  one  term,  n  being  i,  and  that  would 
contain  /  undifferentiated,  because  /  is  zero.  Then  there  will 
remain  two  couples ;  viz.,  terms  2  and  4,  and  3  and  7.  The 
first,  since  I—  i  and  ?i—i,  becomes 

\{2uA,2i'yfY,  (18) 

In  the  last  couple  /  =  o,  ;2  =  2,  and  it  becomes 

{uA,uyt  +  2{uAyty', 

and  rejecting  the  first  term,  we  have 

2{uAyty.  (19) 

Now  collecting  the  terms  from  (16),  (18),  (19),  and  the  first 
and  fifth  of  (17),  the  result  can  be  written  thus: 

1=  {[u'A,  -  4i/''A,+  2uA,ii'  +  2{uA,uyyY  +  {u'Ajy 

=  {B,ty  +  {Bjr,  (20) 

and  this  by  immediate  integration  gives  (12) ;  and 

B,  =  tt'A,  -  4u"A,^  2uA^u"  +  2{uA,tiy  (21) 

and 

B,  =  ti'A,.  (22) 


JACOBPS   THEOREM.  215 

168.  We  may  now  integrate  (11)  by  parts,  thus: 
SU^  1  ritdx  \^^AbJ-\-  {Bjy  \   -  -»  I  Bj'+{Bj'y  \ 

2«^^o  2    L  ^  )   1       2    (  )  0 


But 

If,  therefore,  we  suppose  u  to  be  so  taken  as  not  to  vanish  at 
either  hmit,  t^  and  t^  must  vanish,  and  we  shall  have 

SU=-  \£l'  \bj'^  {B^  t")'  }  t'dx,  (24) 

But  we  see  at  once  that  in  this  case  the  terms  of  the  second 
order  require  still  further  transformation,  as  they  are  not  yet 
in  a  quadratic  form ;  and  to  this  we  now  proceed. 

(69.  Let  Va  be  such  a  quantity  as  will  satisfy  the  differen- 
tial equation 

B,v'a-^{Bya)'  =  0,  (25) 

Then  by  putting  iia  for  v' a,  we  have 

B,Ua-\-{B.u'a)'  =  o,  (26) 

Assuming  for  the  present  that  Va  and  consequently  Ua  can  be 
determined,  (24)  can  be  still  further  transformed.  For  we  see 
from  that  equation  that  if  Ua  were  an  admissible  value  of  f 


2l6  CALCULUS   OF  VARIATIONS. 

throughout  the  limits,  ^^  would  to  the  second  order  reduce 
to  zero.  But  whatever  be  the  value  of  t' ,  we  may  certainly 
represent  it  by  Uata^  and  (24)  will  then  become 

^U^  -  i/J'  j  B^Uata^  \.BlUata)'^;  |  Uatadx 
I     /'^i 

where  I^dx  is,  as  we  shall  show, immediately  integrable  by  the 
note  to  Lemma  II.,  its  integral  taking  the  form 

fl,dx=Cj'a.  (28) 

(70.  To  find  C^,  multiply  (26)  by  tiata  and  subtract  from 
the  value  of  /^  in  (27).     Then  we  shall  have 

I,  =  Ua  [BlUa  taYy  -  Ua  ta{B,  U  ^' ,    '  (29) 

But 

Ua  \  BlUa  taY  V=\  tla  Blu^  ta)']'  -  u' a  B^tla  t^)' 

and 
Whence 

Ua  \  BlUa  ta)'  ]' =  {Ua  B,  U^  t' a)'  +  {Ua  B,  u' ^  4)' 

U  d  Jd^  11(1 1  fi  21  d  x5g  U  0^  1(1, 

Now  since  all  the  terms  in  /,  which  contain  4  undifferentiated 
must  cancel,  we  reject  the  last  term  and  also  the  couple  2  and 
3,  because,  as  ;/  =  i,  they  could  be  united  into  one  term  which 
would,  as  /  =  o,  contain  ta  undifferentiated.  For  the  same 
reason  the  second  term  in  (29)  is  rejected,  and  we  have 


JACOBVS   THEOREM.  21/ 


and 


where 


J  I,  dx  =  Q  t'. 


C,  =  B,  tc\,  (30) 


171.    Resuming (27),  dU  can  now  be  integrated  by  parts, 
thus : 


I    /*^i 


dx 


2^x0 


=  -'-  {C,t\t}  +1  [Cj'at^  ^\£'Cj'\dx.  (31) 

The  following  equations  will  also  hold  : 

Sy  , iiSy'  —  dyti'  f ii6y'  —  dyu' 


21, 


Now  since  u  does  not  vanish  at  either  limit,  and  dy  and  Sy' 
vanish  at  both,  it  is  evident,  from  the  above  value  of  t' ,  that  // 
and  t^  will  become  zero,  which  will  cause  /«  to  vanish  at  the 
limits.  Then  putting  for  Ua  its  value  v' a,  and  for  C^^  the  value 
obtained  by  referring  to  equations  (30)   (22),  and  (9),  we  have 

I  r^^  [  t'  y 

^^  =  Wxo  ^^v^^'^^'a  -7-  ^^'  (32) 


172-  We  must  now  determine  the  form  of  the  quantity  Va, 
and  for  this  purpose  we  must  evidently  solve  (25).  Now  by 
comparing  this  equation  with  (12),  we  see  that  Va  is  what  / 

must  become  in  order  to  render  /     Idx  zero  ;  that  is,  to  ren- 

der  /  zero.  But  /  =  tiSM,  which  will  at  once  appear  if,  in  the 
final  value  of  (^ J/ given  in  Art.  164,  we  write  Sy  =  tit,  Sy'  —  {tit)' 
and  Sy'^  =  {^i^Y,  which  will  in  no  way  restrict  the  values  of  the 


2l8  CALCULUS  OF  VARLATLONS. 

variations.  Hence,  since  u  does  not  vanish,  we  must,  when  /is 
zero,  have  ^M  zero.  Now  we  already  know  that  this  condi- 
tion will  be  satisfied  by  making- 

=  ar^-^hr^-\-cr,^-dr,,  '  (33) 

and  this  condition  can,  since  A,  A^  and  A^  are  not  in  our 
power,  be  satisfied  in  no  other  way.  For  the  integration  of 
the  equation  M  —  o  gives  jr  as  a  function  of  x  and  certain  con- 
stants, the  form  of  the  function  being  determined,  and  the 
values  of  these  constants  only  being  undetermined.  There- 
fore, since  x  does  not  receive  any  variation,  any  change  which 
cannot  be  produced  in  7  by  varying  the  constants  would  cause 
some  change  in  the  form  of  the  function,  and  hence  y,  when 
thus  changed,  could  no  longer  satisf}^  the  equation  J/  =:  o, 
which  it  must  do  in  order  that  dM  may  vanish.  This  reason- 
ing is  evidently  applicable  whatever  be  the  order  of  M. 

Now  it  is  evident  that  we  can  cause  the  second  member  of 
(33),  which  we  know  to  represent  the  most  general  form  of 
II,  to  assume  various  values  for  the  same  value  of  x  by  various 
determinations  of  the  arbitrary  constants  a,  b,  etc.  Let  ti  and 
V  be  any  two  such  values,  so  that  we  may  write 

u^a,r,  +  a^r^-\-  a^  r,  +  a,  r,,  (34) 

v=.b,r,-\-  b,  r,  -\-b,r,  +  b,  r,.  (35) 

But  since  Sy  —  tit,  if  we  make  /  =  — ,  (^J/  will  become  v,  and 

the  equation  dM  —  o  will  be  satisfied,  as  will  also  the  equa- 
tion /  =  o.  Moreover,  this  is  the  only  solution ;  since,  by  suit- 
ably determining  the  constants  in  v,  -  can  be  made  to  equal 

li 


JACOB!' S   THEOREM.  219 

any  value  of  t  which  will  render  /  zero,  and  therefore  every 
value  which  will  render   /  I dx  zero. 

173.  But  the  value  of  v^  is  not  yet  fully  determined.  For 
although,  by  substituting  -  for  t,  we  shall  render  /  zero  what- 
ever be  the  system  of  arbitrary  constants  employed  in  v,  we 
shall  not,  by  such  a  substitution,  necessarily  satisfy  (25).  Be- 
cause when  /  vanishes  independently  of  any  particular  value 

of  V,  J  Idx  is  merely  a  constant.     Hence  all  that  we  can  say 

is  that  the  relation  Va—  -  will  render  the  second  member  of 

u 

(25)  a  constant.  Moreover,  it  is  the  only  relation  which  will 
render  it  a  constant,  because  it  is  the  only  value  of  t  Avhich 
will  cause  /  to  vanish.  Hence,  since  zero  is  a  constant,  if  any 
real  value  of  Va  exist,  it  must  be  capable  of  being  expressed  in 

the  form  -;   only  t;ie  eight  constants,  a^,  etc.,  b^,  etc.,  must 

u 

be  so  related  as  to  satisfy  (25). 

One  of  these  relations  will  immediately  appear.  For,  ex- 
amining (25),  we  see  that  it  is  a  differential  equation  of  the 
third  order  in  Va ;  and  hence  by  integration  we  should  obtain 
Va  as  a  function  involving  not  more  than  three  perfectly  arbi- 
trary constants  of  integration.  If,  however,  we  understand 
only  by  u  and  v  any  two  quantities  of  the  form  given  in  (34) 

and  (35)  in  which  the  eight  constants  are  so  related  that  -, 

when  put  for  t,  will  satisfy  (25),  which  relation  must  cause  the 

V 

constants  to  be  so  combined  that  -  may  contam  not  more 

n 

than  three  arbitrary  constants,  then  we  may  write 

v^  =  -^  (36) 


220  CALCULUS  OF  VARIATIONS. 

1 74-.  Although  this  relation  between  the  constants  was 
noticed  by  Jacobi,  many  subsequent  writers  have  fallen  into 
the  error  of  supposing  that  they  are  entirely  independent,  and 
have  thus  rendered  this  portion  of  their  explanation  untrust- 
worthy. Among  these  writers  is  M.  Delaunay,  who  was  fol- 
lowed by  Prof.  Jellett.  The  latter,  on  page  95,  makes  a  state- 
ment which  would  with  our  notation  be  equivalent  to  saying 
that  whatever  value  of  t  will  make  /  vanish,  will  also  render 

/  Idx  zero,  which  is  manifestly  untrue. 

175.  We  may  now  proceed  to  the  final  transformation  of 
the  value  of  (^t/ given  in  (32).     We  have,  from  (36), 


,        uv  —  vu 

Vn.  = ;: 


u  u  u 

Therefore 


t'  uSy'  —  dyu' 


and 


/  f  Y_  {uv'  -  V7t')  {u^y'  -  dyuj-  (7iSy  -  dyti')  {uv'  -  vit')' 

But 

{uv'  —  vuy  =  uv"  —  vu"  y 

and 

(^^,Sy'  —  dyu')'  =  u^y"  -  Syu". 

Substituting  these  values  in  (32),  reducing,  and  factoring  with 
reference  to  (3j/,  dy'  and  ^y" ,  we  finally  obtain 


[  uv'  —  vu'  f 


JACOBrS    THEOREM.  221 

From  this  equation  we  see  that  to  render  U  a  maximum 
or  a  minimum,  ayy  must  be  of  invariable  sign,  and  should  also 
remain  finite  throughout  the. range  of  integration,  and  not 
vanish  permanently.  If  these  conditions  be  fulfilled,  it  is 
necessary  also  that  the  second  factor  of  (37)  should  not  per- 
manently vanish,  and  it  ought  also  to  remain  always  finite. 
The  first  condition  will  always  be  satisfied.  For  if  in  any 
case  it  were  not,  we  would  have 


SU=\£'6M6ydx  =  o;  (38) 


and  since  every  element  of  this  integral  must  have  the  same 
sign  as  ayy,  which  is  invariable,  (38)  can  only  be  satisfied  by 
making  dM  dy  constant.  But  since  Sy^  and  dy,^  are  zero,  this 
constant  must  be  zero  also,  which  would  render  it  necessary 
that  SM  should  vanish.  But  this,  as  we  have  shown,  would  not 
happen  unless  u  or  ku  be  an  admissible  value  of  Sy ;  and  since, 
as  explained  in  Art.  165,  we  assume  in  (37)  that  such  is  not 
the  case,  it  is  evident  that  the  factor  in  question  cannot  perma- 
nently vanish. 

Hence  we  see  that  if  Uy-y  be  of  invariable  sign,  while  SU 
cannot  be  made  to  vanish  by  the  use  of  u  or  ku,  as  indicated 
in  Art.  165,  neither  can  it  be  made  to  vanish  by  any  other 
mode  of  varying  j/,  y  a,ndy'\  To  satisfy  the  second  condition  it 
is  necessary  that  the  denominator  in  (37)  shall  not  vanish,  and 
that  the  coefficients  of  Sy  and  dy  in  the  numerator  shall  both 
remain  finite.  That  is,  we  must  be  able  to  so  determine  the 
constants  that  uv'  —  vu'  may  not  vanish,  while  u,  u\  u" ,  v, 
v'  and  v"  must  at  the  same  time  remain  finite.  But  before  we 
can  examine  these  conditions,  we  must  be  able  to  express 
these  coefficients  of  Sy,  Sy'  and  Sy"  as  functions  of  x,  and  per- 
fectly arbitrary  constants,  and  we  shall  next  consider  how 
this  may  be  effected. 


(39) 


222  CALCULUS   OF    VARIATIONS. 

176.  Now  from  (34)  and  (35)  we  have 

u  =  a,r,    -\- a^r^    -\- a,r^    +  (t,r„  ] 
u'  =  a,r/  +  a,r,'  +  a,r/  +  a,r/, 

V  =  b^r,    +  b^r^    +  b^r^    +  b,r,, 

v'^  b^r;  J^b^r:  +  b,r:  -\-b,r:, 

W'=  b,r:  +  b,r:+b,r:+b,r:.] 

As  we  wish  to  substitute  these  quantities  in  the  various  parts 
of  the  second  member  of  (37),  we  can  avoid  tedious  multiplica- 
tions and  exhibit  the  results  more  explicitly  by  the  use  of  de- 
terminants.    For  (37)  may  evidently  be  written 


dU 


1  T"^ 


2t/a-o 


dx 


I  r^^^ 


2t/Xo 


LySy-Ly,S/  +  Ly.Sy"\ 


where 


Ly»  — 


Uy       u 


Vs     V 


Ly» 


(5j/,      dy\  dy" 

u,        u',  u" 

v..        v\  v" 
u' ,     u' 


j  dxy 


Ly    


LyI    — 


u,    u 


V,     V 


(40) 


(41) 


Now  for  convenience  we  shall  denote  any  determinant  of  the 
second  order  containing  two  ^'s  and  two  ^'s  by  the  numerical 
suffixes  of  its  first  element,  and  similarly  determinants  with 
respect  to  r,  r' ,  etc.,  will  be  denoted  by  the  numerical  suffixes 


JACOBFS   THEOREM.  223 

of  their  first  elements,  together  with  the  accents  of  r.  Then, 
since  u' ,  tt",  v'  and  v"  have  the  forms  given  in  (39),  while  Ly  is 
a  determinant  of  these  quantities,  we  can,  by  a  well-known 
principle  of  the  subject,  at  once  exhibit  Ly  thus : 


Ly=:i2'i'2"^iyi'7;'-\-  14. iV 

+  23.2Y  +  24.2r  +  34'3V;     (42) 
and  in  like  manner  we  obtain 

Zj,^  =  12.  i2''+i3.i3'^+i4- 14^^+23. 23''+24. 24^^+34. 34^ 
Zy/,=  i2. 12^  +  13. 13^  +  14- 14' +23. 23^+24. 24' +34. 34^ 


(43) 


Hence  if  we  regard  the  determinants  12,  13,  14.  23,  24,  and  34 
as  new  constants,  we  see  that  the  eight  constants  in  u  and  v 
have  so  combined  as  to  leave  but  six  in  equation  (40).  If  now 
we  divide  Z^,  Ly>  and  Lyn  by  one  of  these  constants,  as  12,  and 
denote  the  respective  quotients  hy  My,  My'  zw^  Myn  ,wq  may, 
without  altering  the  value  of  equation  (40),  substitute  these 
quantities  for  Ly,  Ly>  and  Lyn.  Hence  we  require  only  to 
determme  the  forms  of  these  quantities.     But  if  we  write 

a^'X        b^l%        c^X        d='A        e^^X{AA) 

12  12  12  12  12    ^      ^ 

then 

My  =  1^2'^+  a  1^3'^+  d  I V+  ^2^3^^+  d2r+  e  3V, ' 
JZ,.=  i2'^+^i3^^  +  ^i4''  +  ^23''+^24^'  +  r34^    I     (45) 
Myn=  12'  -\-a  13' +  3  14' +^23'+ ^24^ +^34^   ^ 

We  have  now  but  five  constants  to  consider,  and  the  last 
of  these  may  be  expressed  in  terms  of  the  other  four.  For 
we  have 

12.34  +  23.14-  13.24  =  0,  (46) 


224  CALCULUS  OF    VARIATIONS. 

an  equation  which  will  be  found  upon  trial  to  be  identically 
true.     Hence 


and 


34   i_?i.H_  13.24  _ 
12"*"  12  12       12   12 

e-\-cb 

e  z=  ad  —  be  = 

a,    b 
c,    d 

ad 


(47) 


which  value  being  substituted  in  (45)  will  give  My,  My>  and 
Myi>  as  functions  of  four  constants  only. 

Our  reasoning  thus  far  would  hold  even  were  the  eight 
constants  which  enter  u  and  v  entirely  unrestricted.  But 
since  these  constants  must  have  such  mutual  relations  as  will 

satisfy  equation  (25),  where  we  now  know  that  Va  is  put  for  -, 

u 

the  four  remaining  constants  must  also  be  subject  to  some 
restriction,  or  conditioning  equation,  which  will  enable  us  to 
express  My,  My>  and  My»  as  functions  of  not  more  than  three 
perfectly  arbitrary  constants.  But  to  determine  this  last  rela- 
tion in  any  particular  case  it  will  be  convenient  to  present 
equation  (25)  under  another  form,  and  this  we  now  proceed 
to  do. 

177»  Assume  the  equations 

Az  +  {A,z')'  +  {A,zT  =f 

Au+{A,uy+{Ayy'  =r 


and 


Then 

u/-zF=uAz-zAu+u{A,  z')'—z[A,u')' 

-\-n{^A,z")" -z{A,u")"^i-^k.  (48) 


JACOBrS   THEOREM.  225 

Now 

u(A^z')'  =  {uA.zy  —  u'A^z 
and 

zi^A,  u'Y  =  {zAyy  —  z'A,  u'. 

Whence 

i  =  A^(uz'  —  zu')'.  (49) 

Also, 

u(^A,z")"  =z  {uAXY'  -  2{u'A,z"Y-\-u''A,z"; 

and  developing  the  remaining  term  in  like  manner,  and  sub- 
tracting, we  have 

k  =  \A,{uz''  -  ztOY'  -  2\A,{u'z"  -  z'u")Y.  (50) 

But  since  the  second  members  of  (49)  and  (50)  are  integrable 
once,  if  we  add  these  equations,  obtaining  thereby  the  value 
of  uf  —  zF,  and  then  integrate,  we  shall  have 

J  \uf—  zF\dx  =  A,{uz'  —  zic')  —  2AJ^ti'z"  —  z'u") 

+  lA,{uz--zinY.  (51) 

Now  put  ^y  for  z,  and  let  2i  be  such  a  value  of  z  or  Sy  as  will 
render  F  zero.  Then  the  second  term  will  disappear  from 
the  first  member  of  (51),  and  the  remaining  term  will  become 

/  Idx;  and  we  shall  have 

f/dx  =  A,{udy  -  6yu')  -  2A,(u'dy'  -  d/?/') 

+  \A,{uSy'  -  Syzc")  Y  =  B,f  +  {B,  fj.  (52) 


t=^-l- 


But  since  t  —^-^  and  Va  =  -,  we  have  only  to  chansre  ^y  into 
u  u 


226  CALCULUS  OF    VARIATIONS. 

V  in  order  to  cause  /  to  become  Va-  Hence,  finally,  (25)  may 
be  written 

ASuv'  -  vu')  -  2A,{u'v'''  -  v'u")  +  \AJ^uv"  -  vu")\'  =  o  ;    (53) 

and  as  we  may  divide  by  any  constant,  we  may  write,  as  the 
final  conditioning  equation, 

A,   My.   -    2A,  My   +    {A,  My)'     =    Q.     ■  (54) 

It  also  appears  by  differentiation  that 

Lyi  =  L'y»  and  L'y>  =z  Ly-{-  uv'"  —  vu'"  :=  Ly-\-  Lx, 

where  Lx  is  exhibited  by  determinants  in  the  same  manner  as 
the  other  Z's.  Hence,  dividing  these  equations,  as  before,  by 
the  determinant  constant  12,  we  have 

My.  =r     M'y.,  M'y.   =   My  +   M^,  ) 

[  (55) 
M,=  i2"'-\-ai3".'+di4'''  +  c23"'  +  d24'''+e34''''  ) 

It  is  evident,  however,  that  in  order  to  apply  equations 
(54)  and  (55)  to  the  reduction  of  the  constants,  we  must  deter- 
mine the  particular  forms  which  are  assumed  by  A^,  A^,  r^,  r^, 
r^  and  r^,  which  cannot  be  done  so  long  as  the  problem  re- 
mains wholly  general. 

178.  The  following  example  is  presented  merely  as  a  means 
of  illustrating  the  preceding  discussion. 

Problem  XXXI. 

It  is  required  to  apply  Jacobis  Theorem  to  Prob.  V. 

Here  ay>y>  ~  2,  so  that  we  have  next  to  consider  whether 
the  terms  of  the  second  order  can  be  made  to  vanish  by  the 


J  A  GOBI'S    THEOREM. 


227 


use  of  u  or  ku.     Now  the  general  solution,  equation  (6),  Art. 
42,  may  be  written 


Hence  we  have  the  following  equations : 

r^  —  x\  r^  —x\  r^  =  x,  r,  =  i, 
^/  =?>^\  r^  =^2x,  r/  =1,  rl  =  o, 
r/'  =  6xy 


(0 


r/'  =  2,         r/'  =  o,       r/^    =  o, 


^2  =0,  r,  =0,  r, 
u  =  a^x^  -\-  a^x^  -{-  a^x  -\-  a^, 
u'  —  la^ x"^  -\-2a^x-\-  d. 


o. 


(2) 


(3) 


Now  if  the  constants  in  u  can  be  so  taken  that  u  and  u'  shall 
vanish  twice  or  more,  simultaneously,  within  the  limits  of  in- 
tegration, the  terms  of  the  second  order  can  be  made  to  vanish 
by  the  use  of  u,  and  we  have  in  general  neither  a  maximum 
nor  a  minimum. 

Now  if  u  and  u'  can  satisfy  these  conditions,  let  x^  and  x^ 
be  two  values  of  x  for  which  they  vanish  Simultaneously. 
Then  we  must  have 


<  +  ^^,'  +  ^^,  +  -'-  =  o, 


«1 


*==+^^/+^^3  +  --    =    0, 


^1 


x:  +  — ?  ;ir,  +  —?  =  o, 
3^1  3^1 


3    I    2a^        .      a^ 
^3  +  —  -^3  H — -'  =  o. 
3^1  3^1 


(4) 

(5) 
(6) 

(7) 


228  CALCULUS  OF  VARIATIONS. 

Subtracting  (7)  from  (6),  and  (5)  from  (4),  and  dividing  by 
^%~  ^Tg,  we  have 

■^.  +  -^3  + -7-0,  (8) 

<  +  ^,  ^s  +  <  +  "  (^,  +  ^,)  +  -'-  =  O.  (9) 

Substituting  in  (9)  the  value  of  --from  (8),  we  have 

x;  +  -^2  -^3  +  ^"  -  I  {-^.  +  -^3)  (-^2  +-^3)  +  --  =  o 


2X„X^ 


x:    ,  a 


^  +  ?-  Oo) 


2  2        <^j 

Substituting  in  (6)  the  values  of  — '  and  — ^  from  (8)  and  (10), 
we  have,  after  reducing, 

X^  —  2X^  X^  +  X^  —  O. 

Hence  x^  and  x^  cannot  be  different  values  of  x^  and  the  terms 
of  the  second  order  cannot  be  made  to  vanish  by  the  use  of  u  . 
But  since,  as  we  have  seen  in  Art.  175,  these  terms  can  be  made 
to  vanish  by  no  other  mode  of  varying  j,  we  are  sure  of  a 
minimum,  unless,  indeed,  we  cannot  prevent  My  or  Myf  from 
becoming  infinite,  or  My"  from  vanishing  within  the  range  of 
integration ;  and  these  points  we  shall  next  consider. 

179.  Finding,  by  the  use  of  equations  (2),  the  values  oi 
My,  My,  and  My  in  equations  (45),  Art.  176,  and  also  that  of 
Mx  in  equations  (55),  Art.  177,  we  shall  obtain 


Myu  =  —  X*  —  2ax^  —  ibx''  —  cx'^  —  2dx  —  e, 
My>  =  —  4x^  —  6ax^  —  6bx  —  2cx  —  2d, 
My  —  —  6x'  —  6x    —  2c, 
M^  =  —  6x'  —  6ax  —  6b. 


(II) 


JACOBVS   THEOREM.  229 

Now  since  ayuy,,  =  2,  we  see  from  equations  (9),  Art.  164,  that 
A^—o  and  A„  =  2.  Hence,  in  this  case,  equation  (54),  Art. 
177,  becomes 

-  4My  +  2 J/V  ^  o  =  -  2My  +  2 J/^, 

as  will  appear  from  equations  (55)  of  the  same  article.  Equat- 
mg  the  values  of  My  and  M^,  we  have  c  =  ^l?.  Now  taking 
the  value  of  e  from  equation  (47),  Art.  176,  and  then  substi- 
tuting in  the  first  of  equations  (11)  3^  for  c,  we  shall  have,  after 
changing  signs, 

—  My>^  =  x'  -{-  2ax^  +  ddx""  +  ^^^  -\-  ad  —  3<^'',  (12) 

—  My'  =  4x^  -\-  6ax'^  4"  1 2^^  +  2dy  (13) 

—  My   =  6x'  +  6ax  +  6b.  (14) 

It  therefore  at  once  appears  that  neither  My^  nor  My  can  be- 
come infinite  so  long  as  a^  b,  d  and  x  remain  finite.  We  can 
also  evidently  choose  these  constants  in  such  a  manner  that 
Myn  shall  not  vanish  within  the  limits  of  integration.  For 
suppose,  for  example,  that  we  make  both  a  and  d  zero.  Then 
to  render  the  equation 

M  ,t  ^* 

3  2 

possible,  we  must  have 

Hence  if  we  assume  b  greater  or  less  than  this  value  can  be- 
come within  the  hmits  of  integration,  and  also  make  a  and  d 
zero,  we  shall  secure  that  Myi,  will  not  vanish  at  all  as  we  pass 
from  x^  to  ,r,;  and  therefore,  as  all  the  requisite  conditions  can 
be  satisfied,  we  are  in  this  case  sure  of  a  minimum. 


230  CALCULUS  OF  VARIATIONS. 

(80.  We  have,  then,  the  following  general  method  of  ap- 
plying the  theorem  of  Jacobi  in  this  case. 

First  find  whether  ayuyu  remains  finite,  does  not  vanish  per- 
manently, and  is  of  invariable  sign  throughout  the  range  of 
integration ;  because  if  these  conditions  be  not  fulfilled  there 
is  no  need  of  any  further  investigation.  But  if  they  be  satis- 
fied, next  try  whether  dU  can  be  made  to  vanish  by  th© 
use  of  u. 

For  this  purpose  we  write 


and 


a^  a^  a^ 


u'  =  r/  +  --i  r/  -t-  --  r,'  +  -^  r/. 
a,  a,  a. 


Then  if  SU  c?in  be  made  to  vanish  by  the  use  of  u,  the  follow- 
ing equations  must  be  possible  : 

u^^  —  o,         u^  =  0,         u^  =  o,         u^  =  o, 

where  neither  x^  nor  :r^  must  fall  without  the  limits  of  integra- 
tion.    To  determine  the  possibility  of  these  equations  we  first 

eliminate  between  them  the  constants  — ,  ~  and  --,  by  which 

we  shall  arrive  at  an  equation  containing  only  x^,  x^,  and  such 
constants  as  enter  y  in  the  equation  of  the  curve  represented 
by  the  solution.  It  may  then  happen,  as  in  the  preceding  exam- 
ple, that  we  can  determine  the  possibility  of  satisfying  this 
equation  within  the  limits  of  integration.  Or,  if  necessary, 
we  can,  by  using  the  values  of  y,  y\  etc.,  obtained  from  the 
equation  of  the  curve,  eliminate  all  constants  but  numbers, 
thus  securing  a  numerical  equation  between  x^,  x^,  y^,  j/g,  j/, 
etc.,  which  it  must  be  possible  to  satisfy  within  the  limits  of 
integration. 


JACOBPS   THEOREM.  23 1 

If,  then,  it  be  possible  to  satisfy  this  equation,  we  infer,  as  in 
Case  I,  that  we  have  neither  a  maximum  nor  a  minimum.  But 
if  the  hmiting  values  of  u  and  u'  cannot  be  made  to  vanish 
simultaneously,  we  may  assume  that  we  have  a  maximum  or 
a  minimum  according  as  ay,>yn  is  negative  or  positive. 

This  assumption  will,  however,  be  subject  to  any  doubt 
arising  from  the  possibility  that  we  may  not  be  able  by  any 
selection  of  constants  to  prevent  My  or  My,  from  becoming 
infinite,  or  Myn  from  vanishing  for  some  value  of  x  within  the 
limits  of  integration,  thus  rendering  the  corresponding  ele- 
ment oi  ^U  infinite.  To  dispose  of  this  doubt,  we  must,  in 
the  next  place,  actually  find  the  quantities  My,  My,  and  My,,,  and 
possibly  Mx',  as  functions  of  x,  and  but  three  arbitrary  con- 
stants, any  constants  which  may  enter  r^,  etc.,  not  bemg 
reckoned.  But  this  latter  step,  which  will  usually  involve 
difficulty,  may  in  general  be  omitted. 

181.  Some  exceptions  also  occur  in  the  treatment  of  this 
case  which  are  similar  to  those  mentioned  under  Case  i  (see 
Art.  157).  We  shall,  however,  merely  indicate  these  excep- 
tions, the  discovery  of  which  appears  to  be  due  likewise  to 
Spitzer.     (See  Todhunter's  History  of  Variations,  Art.  276.) 

Suppose,  first,  ay„y„  to  become  zero.  Then  it  is  shown  that 
in  order  that^may  become  a  maximum  or  a  minimum,  A^  must 
have  respectively  a  positive  or  negative  sign  throughout  the 
range  of  integration. 

Suppose,  in  the  second  place,  that  we  have  ay„yn  zero,  and 
also  A^  zero,  A  and  A^  having  the  values  given  in  equations 
(9),  Art.  164.  Then  it  is  shown  that  in  order  that  U  may 
become  a  maximum  or  a  minimum,  A  must  be  respectively 
negative  or  positive  throughout  the  range  of  mtegration. 
Moreover,  in  this  case,  as  in  Case  i.  Art.  157,  we  shall  find 
that  the  equation  J/=  o  will  not  be  a  differential  equation  in 
y,  but  merely  an  ordinary  algebraic  equation,  and  that  there- 
fore y  will,  without  integration,  be  determined  as  a  function  of 


232  CALCULUS  OF  VARIATLONS. 

X,  Hence,  geometrically,  there  will  be  no  solution  unless  the 
limiting  values  of  y  and  y'  happen  to  satisfy  the  equation  of  a 
particular  curve  or  class  of  curves. 

Suppose,  lastly,  that  ay^yf,^  A^  and  A   become   severally 

zero.    Then,  as  in  Case  2,  Art.  157,  the  equation  U  —  J     Vdx  is 

capable  of  being  integrated,  and  therefore  the  maximum  or 
minimum  state  of  U  must,  if  at  all,  be  found  by  the  differential 
calculus ;  and  if  the  limiting  values  of  x,  y  and  y'  be  fixed,  U 
will  have  neither  a  maximum  nor  a  minimum  state. 

It  is  evident  that  in  all  these  cases  V  contains  merely  the 
first  power  of  y" ,  and  they  are,  therefore,  like  those  in  Art. 
157,  only  examples  of  Exception  2,  Art.  51. 

(82.  As  the  most  general  case  of  Jacobi's  Theorem  is  pre- 
cisely analogous  to  that  already  explained,  and  as  it  is  rather  of 
analytical  than  practical  importance,  we  shall  merely  indicate 
the  method  of  effecting  the  required  transformation. 

Case  3. 

Let  U  =  J     Vdx,  where  Fis  any  function  of  x,y,  y\  .... 

y^'^^.  Then  the  general  solution  J/  =  o  will  usually  give  y  as 
a  function  of  x  and  2n  arbitrary  constants  of  integration,  and 
these  2n  arbitrary  constants  must  be  so  determ.ined  as  to  sat- 
isfy the  conditions  at  the  limits,  where  we  shall  always  suppose 
the  limiting  values  of  x,  y,  y\  .  .  .  .  y^''^  -  ^)  to  be  assigned. 

Now,  as  before,  since  these  conditions  hold  at  the  limits, 
and  the  terms  of  the  first  order  must  vanish,  we  may  write 


and  may  then,  by  Lemma  L,  put  ^J/ under  the  form 


6M=Ady-{-  [A,Sy    +etc +[AnSj 


/(») 


JACOBVS    THEOREM.  233 

We  shall  also  have,  in  this  case, 

u  =  A,r,^  A,  r,  +  etc: +  ^sn^sw 

Hence,  by  changing  (^J/  into  ul,  and  integrating  by  parts  with 
the  aid  of  Note  to  Lemma  II.,  we  shall  obtain  a  result  which 
may  be  written 

-  ^X^  1  ^^^'+{bJ')\  etc.  ....  +[Bj''f~''  }  /V^. 

Then,  as  formerly,  putting  Ua^a  for  /',  and  integrating  again 
by  parts,  we  have 

d[/  = 

jfC  {  ^^  ^'^  +  (^^  ^'«)  +  ^^^ +  (^^  ta^--'^^'"^^  \  t'adx. 

In  this  equation  we  may  change  t' a  into  ?/64,  where  ?/&  =  zv\, 
and  wi)  is  a  quantity  which  satisfies  the  differential  equation 

C,zv\-\-  [c,w\'^-\-  etc +  ((f^-^ftC^-DJ^''''^^  o. 

Making  this  change,  and  integrating  by  parts,  as  before,  we 
have 


^U=--J^;\Dj',^[Dj\)^^tc. 


!  \  (n  -  3)  ) 

Continuing  this  process  n  times,  we  shall  evidently  arrive  at 
a  result  which  may  be  written 


2  ^^0 


234  CALCULUS  OF    VARIATIONS. 

the  positive  or  negative  sign  being  used  according  as  n  is 
even  or  odd. 

Now  it  is  evident,  from  the  mode  in  which  the  integration  is 

effected,  that  H must  equal  A^ ?/ u^a u\  .  .  .  .,  and  An=  ±  — -1, 

the  positive  or  negative  sign  being  used  according  as  n  is  even 
or  odd,  as  will  at  once  appear  if  w^e  form  the  functions  A,  A^y 
etc.,  by  Note  to  Lemma  I. 

(83.  Let  us  next  consider  the  quantities  zi,  zia,  Uh,  etc.     We 
have,  by  the  same  reasoning  as  that  hitherto  employed, 


u  —  a,r,-{-  a^r^  +  etc +  ^su^sn,         i^a  —  'v' 


a> 


Va=  -,         v  =  b,r,-\-  b^r^  +  etc +  b2nr2n 

li 


(0 


But  the  2n  constants  a  and  the  2n  constants  b  are  not  entirely 
independent,  but  must  be  so  related  that  Va  may  satisfy  the 
equation 

B,  v'a  +  [b,  v"}I^  etc +  (^,, vjf^^'^' '^=  o ;  (2) 

that  is,  Va,  when  put  for  t,  must  render  /  Idx  zero. 
The  following  relations  are  also  evidently  true : 

/  =  21 SM,         /j  =  tiaj  I dx,         /j  —  tij)  J  /i  dx,  etc.  (3) 

Now  to  determine  the  nature  of  Uh,  we  see  from  (i)  that  zvi,  is 
a  quantity  which,  being  put  for  /«,  will  render  y  I^dxzQro\  that 

is,  will  render /j  or  ?/«  j  /^;t- zero, will  render  /  Idx  or  udM zero, 

will  render  dM  zero.    But  since,  in  /  Idx,  t'  is  replaced  by  Ua  ta. 


JACOBTS   THEOREM.  235 

f 
ta  —  —  'j  and  in  order  to  render  that  integral  zero,  the  f  in 

the  value  of  ta  just  given  must  now  be  so  restricted  as  to  sat- 
isfy the  equation 

B,  t'+in,  t'^+  etc +[b^  t(-)Y  "  '^  =  o ; 

that  is,  it  must  render  /  Idx  zero,  or  /  zero,  or  m^M  zero,  or 

(SM  zero.    But  always  f  =  — ,  and  we  can  make  SM  vanish 

only  by  making  Sy  equal  to  some  quantity  of  the  general  form 
of  u. 

Assume  ^y  =  w,  where 

w  =  c,r,  +  c,r,  +  etc +  C2n  ^2^. 

This  will  make  ^M  vanish  ;  and  if  the  271  constants  c  and  the 
2n  constants  a  be  suitably  connected,  t,  which  now  equals 

—  or  Wa,  will  also  satisfy  the  equation  /  Idx  =  o,  which  would 

not  necessarily  happen  if  these  constants  were  entirely  inde- 
pendent. 

We  have  now,  as  the  value  of  /«  which  was  required  to 

render   /  Idjt^  zero, 


(?)• 


^a  —  1 — \t- 

- 

ul 


But  it  does  not  follow  that  every  value  of  /«  which  will  render 
J  Idx  zero  will  also  render  /  /^  dx  zero,  and  w^  must  be  such 
a  value  of  /«•     Still  it  is  evident  that  wi^  can  be  of  no  other 


236  CALCULUS  OF    VARIATIONS. 

general  form  than  that  just  given  for  ta\  only,  in  addition  to 
the  relations  already  noticed  between  the  constants,  the  con- 

stants  in  u,  v  and  w  must  be  so  related  that-; — -/may  render 

It } 


/'• 


dx  zero. 


In  a  similar  manner  we  may  determine  Uc  =  z' c,  but  will 
then  be  obliged  ultimately  to  introduce  into  dt/" another  quan- 
tity of  the  form 

z-=d^r,-\-d^r^^  etc +  ^2^  ^2^. 

Moreover,  these  four  sets  of  constants  will  then  be  subjected 
to  three  more  conditions  ;  six  in  all.  For  Zc  must  be  so  taken 
as  to  reduce  to  zero  the  following  expressions : 

fl.dx,         Jl,dx,  fidx,         SM; 

the  last  condition  serving  merely  to  introduce  z,  but  imposing 
no  restriction  upon  its  constants. 

Thus  it  appears  that  each  increase  by  unity  of  it  will  intro- 
duce into  SU  one  more  quantity  like  z,  and  that  each  such 
new  quantity  will  require  one  more  additional  condition  than 
did  its  predecessor,  the  first  condition  being  introduced  by 
the  second  of  these  quantities. 

Now  T  can  always  be  found  in  terms  of  the  preceding 
quantities.     For  we  have 

^  =  — ,        ta  =  ~,        h  =  i^,        etc.  (4) 

U  Ma  Uh 

Whence  w^e  see  that  by  means  of  7^  the  final  value  of  SU\N\Vi 
be  made  to  involve  dy,  Sy\  ....  dy^),  which  should  evidently 
be  the  case. 


JACOBI'S   THEOREM. 


237 


184.  The  analogy  of  the  preceding  cases  would  lead  us  to 
expect  that  when  the  reductions  indicated  in  the  last  article 
are  performed,  c^^will  assume  a  determinant  form;  and  such 
is  the  fact.  This  subject,  and  indeed  the  whole  theorem  of 
Jacobi,  has  been  most  elaborately  discussed  by  Otto  Hesse  in 
a  paper  which  may  be  found  in  the  54th  volume  of  Crelle's 
Mathematical  Journal  ior  1857,  p.  227,  and  we  are  indebted  to 
this  author  for  much  of  the  preceding  discussion,  and  in  par- 
ticular for  that  part  which  exhibits  the  relation  between  the 
constants  and  the  manner  in  which  they  combine  and  reduce. 
We  shall,  however,  here  merely  give  some  of  his  results. 

Let  ti,  V,  w,  .  .  .  .  X  hQ  n  quantities  which,  being  put  for 
dy,  will  severally  render  6M  zqyo.  Then  we  see  from  Art.  183 
that  dU  will  involve  all  these  quantities.     Let 


L  = 


Sy,     d/,  ....  r^J/(«) 
u,       u\  ....    ^^("•> 


X,    x\ x^-^ 


Lyn    


u^      u  ,  .  .  .  .  u 


in  -  1) 


V,      v\ -.X^-1) 


x.x\ 


Xin  -  1) 


L  being  a  determinant  of  the  order  7i-^\,  and  Z^  a  deter- 
minant of  the  order  n.  Then  Hesse  shows  that  dU  will  take 
the  form 

\_   r^^  d'V  I  L 


dU 


dy^^)'  \L 


,yn 


dx. 


It  is  also  evident,  from  Art.  183,  that  the  number  of  the  con- 
ditioning equations  between  the  constants  involved  in  u,  v, 

.  .  .  .  X  must  be  the  sum   14-2  +  3-}-  etc 4-/2—1,  or 

n{n  —  i) 


238  CALCULUS  OF  VARIATIONS. 

We  may  here  collect  a  few  of  these  conditioning  equa- 
tions, the  first  arising  from  u^,,  the  next  two  from  2^5,  and  the 
last  three  from  Uq. 


^,^'a  +  etC +(^^^^Jr.)y^~'^  ^O^ 

C,w\^QtC -|-(|(f^,^,(n-1)y^"'^^0, 

/  \  (71  -  1) 

^,^^a+etC J^\B^^^^n)\^  .^o, 

f  \(n-2) 

C, z\  +  etc ^[Cn ^b^^ - i>j         -  o, 

/  \(n-3) 

A-^'c  +  etC J^[DnZ,^n-2)\  ^o^ 


w 

z 

^  a 

Wa  —  -, 

Za  —  -, 

U 

U 

where 


Some  of  these  relations  are  more  explicitly  exhibited  by  Hesse, 
but,  for  a  reason  which  will  presently  appear,  it  is  unnecessary 
to  go  any  further  into  this  matter. 


Now 


_m 


Zc  =  — ,— »        '^b  =  —r-,        and     Va  =  -' 

Hence  it  appears  that  ^5  is  a  differential  expression  of  the 
first  order,  Zc  of  the  second,  etc. 

186.  The  manner  in  which  the  constants  enter  <^6^  is  similar 
to  that  in  Case  2.     For  L  may  be  written 

Ly  dy+  Ly.  dy  +  etc +  Z^n  dyin\ 


JACOBPS    THEOREM.  239 

where  Ly,  Ly>,  etc.,  are  themselves  determinants  of  the  nth. 
order.  But  if  in  any  of  these  determinants  we  substitute 
the  values  of  its  constituents,  we  know  that  the  determi- 
nant will  become  the  sum  of  products  of  pairs  of  determi- 
nants, each  product  consisting  of  a  determinant  of  the  nth 
order  in  constants,  multiplied  by  a  determinant  of  the  same 
order  in  the  rs  and  their  differential  coefficients,  there  being 
as  many  such  products  as  there  are  combinations  of  2n  num- 
bers, taken  n  in  a,  set,  no  two  determinants,  whether  variable 
or  constant,  being  the  same. 

This  is,  however,  as  far  as  we  can  go.  For  to  show,  in 
general,  how  these  determinant  constants  combine  so  that 
Ly,  Ly>,  ....  Lyr.  may  be  expressed  as  functions  of  x  and  en- 
tirely independent  constants,  is  a  problem  which  has  not  yet 
been  solved.  Now  in  order  that  no  element  of  (5' ^maybe- 
come  infinite,  we  must  be  able  to  so  determine  the  arbitrary 
constants  that  Ly^  shall  not  vanish,  and  that  none  of  the  quan- 
tities Ly,  Ly,y  ....  Ly^  may  become  infinite  within  the  range 
of  integration.  But  the  above  defect  will  prevent  us  from 
determining  whether  or  not  these  conditions  can  be  fulfilled, 
since  it  will  prevent  us  from  obtaining  these  quantities  as  ex- 
plicit functions  of  x  and  entirely  independent  constants. 

186.  After  a  general  discussion,Hesse  considers  successively 
the  cases  in  which  ?/  is  i,  2  and  3.  In  the  latter  case,  the  con- 
stants will  enter  Ly,  Ly>,  ....  Lyn,  in  the  form  of  twenty  deter- 
minant constants  of  the  third  order,  and  the  conditioning  equa- 
tions will  be  three  in  number.  Moreover,  between  these  twenty 
determinants  there  subsist  thirty  identical  equations  analogous 
to  equation  (46),  Art.  176.  Now  by  division,  as  before,  Ly, 
Ly>,  ....  Lyr.  become  My,  My,,  ....  J/^n,  and  these  constants 
may  be  reduced  to  nineteen.  Then  the  three  conditioning 
equations  should  enable  us  to  reduce  them  to  sixteen,  and 
finally  the  thirty  identical  equations  are  of  such  a  character  as 
to  enable  us  to  eliminate  but  ten  more  determinants. 


240  CALCULUS  OF   VARIATLONS. 

Thus  it  will  appear  that  there  remain  not  more  than  six  irre- 
ducible constants.  Hesse  does  not  say  that  these  constants 
are  yet  perfectly  independent,  and  the  author  is  not  prepared 
to  say  more  than  that  they  appear  to  be  so.  For  a  further 
discussion  of  this  subject  the  reader  is  referred  to  the  paper 
in  question. 

187.  We  see  then,  in  general,  that  in  order  that  U  may 
have  a  maximum  or  a  minimum  state,  it  is,  in  the  first  place, 

necessary  that  or  ^^n^n  shall  remain  finite,  not  vanish  per- 

manently, and  be  of  invariable  sign  throughout  the  limits 
which  we  wish  to  consider.  This  principle,  however,  is  not 
due  to  Jacobi,  it  having  been  enunciated  by  Legendre  as 
early  as  the  year  1786.  Still,  the  method  of  discriminating 
maxima  and  minima  given  by  Legendrp  and  Lagrange  was 
defective,  because  it  gave  no  means  of  determining  whether 
some  element  of  c^f/ might  not  become  infinite,  as  it  always 
employed  certain  functions  which  could  not  be  determined. 
(See  Todhunter's  History  of  Variations,  Arts.  5,  199.) 

If  the  conditions  with  regard  to  ay^yr.  indicate  a  maximum 
or  a  minimum,  we  must,  in  the  next  place,  determine  whether 
d^can  be  made  to  vanish  by  the  use  of  u,  since,  if  it  can,  there 
is  no  need  of  Jacobi's  transformation,  and  we  infer  at  once 
that  U  has  neither  a  maximum  nor  a  minimum  state.  To  make 
d^thus  vanish  we  must  be  able  to  satisfy  the  equations 


U^  —  O, 

u,'  =  0,  etc., 

^^(n-l)_o, 

u^  —  0, 

u,'  =  0,  etc., 

^^(n-l)  —  Q^ 

where  neither  x^  nor  x^  must  fall  without  the  limits  of  inte- 
gration. 

Geometrically,  we  may  regard  j  in  any  proposed  solution 
as  the  ordinate  of  a  curve  whose  extremities  are  the  points 
;i:o,  Jo  and  ^i,  j\.     Then  the  proposed  value  off  will  render  U 


JACOBPS    THEOREM.  24 1 

neither  a  maximum  nor  a  minimum,  if  it  be  possible,  by  mak- 
ing infinitesimal  changes  inj,y, . . .  .y^),  to  draw  another  curve 
meeting  the  first  at  the  points  x^,  y^  and  x^,  y^,  and  having  at 
these  points  the  same  values  of  y, ....  y"-  - 1>,  and  also  satisfy- 
ing the  equation  M  ^  o. 

Now  although,  when  the  limiting  values  of  x,  y,  y\  .  .  .  . 
y{n-\)  are  assigned,  all  the  constants  which  enter  the  equation 
of  the  curve  which  satisfies  all  the  conditions  of  the  question 
are  determined,  yet  as  this  determination  is  not  alway  abso- 
lute, allowing  us  a  choice  of  two  or  more  values,  there  will  in 
general  be  more  than  one  such  curve,^as  in  Prob.  VII.,  where 
two  catenaries  can  often  be  drawn,  both  satisfying  the  condi- 
tions of  the  question.  Now  if  such  limits  be  found,  in  passing 
along-  one  of  these  curves,  as  to  render  it  and  another  curve 
coincident  between  these  limits — that  is,  if  the  equation  of  this 
curve  have  one  or  more  pairs  of  equal  roots — dU  to  the  second 
order  can  be  made  to  vanish,  and  we  infer  that  U  has  neither 
a  maximum  nor  a  minimum  state. 

If  we  can  assure  ourselves  that  SU  will  not  vanish,  then  we 
must,  in  the  third  place,  determine  Ly,  Ly>,  ....  Z^.»,  in  order  to 
ascertain  whether  or  not  all  the  elements  oi  dU  remain  finite. 
But  this  point  has  been  already  fully  treated,  and  we  have 
seen  that  this  determination  cannot  always  be  effected.  When 
this  is  the  case,  the  theorem  of  Jacobi  is  practically  subject  to 
the  same  defect  as  existed  in  the  method  of  Legendre  and 
Lagrange.  It  will  appear,  however,  that  by  determining  the 
function  tc,  which  is  always  possible  when  the  complete  inte- 
gral of  the  equation  M=o  can  be  obtained,  and  sometimes 
when  it  cannot,  we  may  frequently  be  able  to  infer  that  U  has 
neither  a  maximum  nor  a  minimum  state,  even  when  dy^yn  is 
always  finite  and  of  invariable  sign  ;  and  this  inference  could 
not  be  drawn  from  the  above-named  method. 

188.  From  the  cases  in  which  n  is  i  and  2,  we  might  natu- 
rally expect  that  some  exceptions  to  the  theorem  of  Jacobi 


242  CALCULUS  OF    VARLATLONS. 

would  arise  wnen  n  is  greater  than  2,  particularly  if  ayr^yx. 
should  happen  to  become  zero  throughout  the  range  of  inte- 
gration ;  and  such  appears  to  be  the  fact.  For  Spitzer  has 
examined  also  the  case  in  which  n  is  3,  and  has  shown  that 
certain  forms  of  Fgive  rise  to  exceptions.  We  subjoin  from 
Todhunter's  History,  Art.  278,  the  following  four  forms  of 
F,  which  the  reader  may  examine  for  himself : 

v=A^,y,y',y")^y"'fa{x,y,y',y"), 

V = /(^,  y,  y')+y"fa  {x,  y,  y')+  \  h{x,  y,  y,  y")  \ ', 

V  =  Ax,y)+y'fa{x,y)+  \f^{x, y, y')\'^  \  fc{x, y, y', /')]', 

V=yAx)+\fa{x,  y)V+  \Mx,  y,  y')V+\fc{.x,  y,  y',  y")}', 

where  /,  /«,  fb  and  fc  are  any  functions  whatever.  Hesse 
does  not  mention  the  existence  of  any  exceptional  cases, 
although  he  had  seen  the  discussion  by  Spitzer. 

It  will  be  observed  that  in  applying  Jacobi's  Theorem  we 
have  always  regarded  the  limiting  values  of  x,  y,  /,..  .  y^-*) 
as  fixed,  thus  rendering  the  discussion  somewhat  restricted. 
But  the  solution  of  the  more  general  problem,  that  in  which 
these  limiting  values  are  also  variable,  if  it  be  at  all  possible, 
has  up  to  the  present  time  baffled  the  skill  of  those  who  have 
attempted  it. 

189.  Before  closing  this  section  we  must  mention  one  point 
with  regard  to  the  terms  of  the  second  order  not  strictly  con- 
nected with  the  theorem  of  Jacobi. 

We  have  already  seen  that  the  simplicity  of  the  form  in  which 
these  terms  appear  is  often  dependent  upon  our  choice  of  the 
independent  variable,  and  it  may  therefore  be  well  to  consider 
particularly  how  the  terms  in  6U,  derived  by  regarding  x  and 
y  successively  as  the  independent  variable,  are  connected,  and 
why  they  are  not  identical. 


CHANGE   OF  INDEPENDENT  VARIABLE.  243 

Assume  the  equation 

u^jydx,  (I) 

where  Fis  any  function  of  x,y,y' , ....  y^),  the  limiting  values 
of  X,  /,  y, .  .  .  .y"'-i)  being  fixed.  Then,  since  both  x  and  y 
are  implicitly,  at  least,  involved  in  U,  we  may  regard  y  as  some 
function  of  x,  and  may  therefore  suppose  it  the  ordinate  of 
some  primitive  curve  for  the  abscissa  x.  Now  by  varying  U 
we  must  pass  to  some  derived  curve,  and  let  Y  or  y  -{-  dy  be- 
come the  ordinate  of  this  curve  for  the  same  abscissa  x. 

Next  taking/  as  the  independent  variable,  and  expressing 
U  in  terms  of  y,  dy,  x  and  its  differential  coefficients  with 
respect  to  j,  equation  (i)  may  be  written 

V.^Jyy',dy,  (2) 

the  limiting  values  of  y,  x,  x\  etc.,  being  also  fixed,  where 

dr 
x'  —  -^,  etc.     Moreover,  ^and  U.  will  be  identical  when  the 
dy 

relations  between  x  and  y  in  (i)  and  (2)  are  the  same  ;  that  is, 

when  y  is  an  ordinate  of  the  same  primitive  curve  in  both  for 

the  same  abscissa.     Varying  (i)  and  (2)  and  transforming  the 

terms  of  the  first  order,  observing  that  the  limiting  values  are 

all  fixed,  we  have 

\-^^'^^s.y^yd^^\s.yd^+s^!'d-^'     (3) 

I'  UA  =Sy'-  ^^  +  J  ,CS  dy  +  £  Tdy,  (4) 

where  brackets  denote  the  entire  increment  which  ^and  U^ 
receive  by  variation,  the  integrals  following  M  denoting  re- 
spectively the  terms  of  the  second  order  and  all  those  of  a 
higher  order,  and  those  following  N  having  a  similar  significa- 


244  CALCULUS   OF  VARIATIONS. 

tion.  Now  supposing  U  and  U^  identical,  the  first  members 
of  (3)  and  (4)  will  become  equal  if  in  (2)  we  so  vary  x  as  to 
obtain  the  same  derived  curve  as  we  did  from  (i)  by  varying 
y ;  and  this  requires  dx  to  have  such  a  value  that  y  may  be  the 
ordinate  of  the  derived  curve  for  the  abscissa  x  -|-  ^x.  Hence, 
by  tracing  along  the  derived  curve  from  the  point  whose  co- 
ordinates are  x  and  Y  to  that  whose  co-ordinates  are  x  -{-  dx 
and  y,  we  see  that 

.     y=  V+  Ydx  +  -  Y'Sx'  +  etc. ; 

and  putting  for  V  its  value  y  +  ^y,  we  find 

dy  =  —  y'dx  -\-  w,  (5) 

S.=  -%  ,     (6) 

y 

where  w  contains, only  terms  of  an  order  higher  than  the  first. 
Now,  since  the  second  members  of  (3)  and  (4)  are  absolutely 
equal,  the  terms  of  the  first  order  in  these  two  members  can- 
not .differ  by  any  term  of  the  first  order.     Hence,  and  from 

(6),  observing  that  dx  =  -^,  we  have,  to  the  first  order, 

MSydx=         NSxdy=:         -Ndydx; 

so  that  N  =  —  M.  Still,  denoting  by  a  the  terms  of  the  first 
order,  and  by  b  those  of  the  second,  in  (3),  and  by  c  and  d  the 
corresponding  terms  in  (4),  we  cannot  say  that  a  and  c  are 
absolutely  equal,  but  they  cannot  differ  by  more  than  some 
term  of  the  second  order,  and  they  will  in  general  differ  by 
such  a  term.  In  like  manner,  a -\-  b  and  c  -{-  d  cannot  differ 
by  any  term  of  the  second  order,  although  they  may  differ  by 
some  term  of  the  third,  and  therefore  b  and  d  may,  and  in 
general  will,  differ  by  a  term  of  the  second  order. 


CHANGE    OF  INDEPENDENT    VARIABLE.  245 

190.  Now  if  ^is  to  be  a  maximum  or  a  minimum,  and  we 
express  it  successively  as  in  (i)  and  (2),-  then  a  and  c  must 
each  vanish,  because  M  and  N  vanish,  and  we  may  then  find 
that  d  is  much  more  simple  than  b ;  and  as  these  terms  must 
now  be  equal  as  far  as  the  second  order,  because  a  and  c  have 
become  zero,  we  conclude  that  b  must  contain  an  expression 
which  adds  nothing  of  the  second  order  to  its  value,  and  that 
this,  by  the  second  method,  becomes  involved  in  c,  thus  leav- 
ing b  in  the  simpler  form  d. 

We  know,  moreover,  that  M  and  N  will  be  entirely  inde- 
pendent of  the  conditions  which  may  be  required  to  hold  at 
the  Hmits,  so  that  the  relation  N  —  —  M  must  hold  whether 
the  hmiting  values  of  x,  y,  /,  etc.,  be  assigned  or  not.  Now 
if  the  limiting  values  of  x  be  fixed,  while  those  of  y  are  vari- 
able, then  if  we  change  the  independent  variable  to  y,  we  may, 
by  regarding  the  limiting  values  of  y  as  fixed  and  those  of  x 
as  variable,  pass  to  the  same  derived  curve  as  by  the  first 
method.  But  the  abscissae  of  the  extreme  points  will  now  be 
x^  -{-  dx^  and  x^  -\-  Sx^,  whereas  they  are  required  to  be  x^  and 
x^  merely.  Hence  to  render  \_^U^  and  [^^]  equal,  we  must 
subtract  from  the  former  the  increment  which  U  would  re- 
ceive in  virtue  of  the  change  in  the  limiting  values  of  x ;  and 
we  know  that  to  the  first  order  this  increment  is 

V^Sx,  -  V,Sx,.  (7) 

Now  when  [SU]  and  \SU^  have  been  made  equal,  as  just 
explained,  it  is  easy  to  find  what  must  be  the  values  of  the  co- 
efficients of  Sx^,  dx^,  6x^',  etc.  For  let  ;/  be  2,  and  x  the  inde- 
pendent variable.  Then,  by  equation  (5),  Art.  36,  the  terms  at 
the  upper  limit  will  be 

(A  -  Q/W.  +  QA\''  (8) 

But  from  (5)  we  have,  to  the  first  order, 

<v  /  //  -X  ,  dSx 

Oy    ^-y     Ox-y'  -—', 


246  CALCULUS  OF  VARLATLONS. 

and  putting  for  dx  its  value  -^,  and  observing  that  — —  =  dx\ 

y  °  dy 

we  have 

dy=:  -y'6x-y"dx'.  (9) 

Substituting  these  values,  (8)  becomes 

\-y:{P,  -  Q:)-yrQ.\Sx-y;^Q,Sx:,  (10) 

and  a  similar  equation  holds  for  the  lower  limit. 

(91.  Two  simple  examples  will  serve  to  illustrate  the  pre- 
ceding discussion. 
First  assume 

U  =£y'd-  =  £  Vdx,     U,  =  jyx'dy  =  £'  K  dy. 
Whence 

^^^^  ^Jy^''  ^^'^y  "^  y^'  ^^~  y^'  ^^~Sl''^y  ^^^^y- 

Now  to  render  {^U]\  equal  to  \pU\  as  far  as  the  first  order,  we 
must  subtract  from  the  former  V^Sx^  —  V^Sx^,  or  y.'^^x^  — 
y^^^^^  which  will  eliminate  all  the  terms  at  the  limits,  as  it 
evidently  should;  and  N  —  —  M.  Still  we  must  recollect 
that  we  have  made  [<^C/]  and  {_^U'^  equal  to  the  first  order 
only. 

As  a  second  example,  let 


U^    r'y''dx:=^    r^Vdx. 


I                             x"                            x'" 
x"         y              x'^'         ^               x'^ 

+ 

J,            x'"       lO;r'V"        isx'" 

•^     ~        ^"   '         x"              x"   ■ 

CHANGE    OF  INDEPENDENT    VARIABLE.  24; 

Then 

[<yC/]  =  -  2j,"'  <5y,  +  2^/'  .Jj/  +  2^/"  Sy,  -  2y:'  Sy: 

Jr£y/-'Sydx+£yy"'dx.  (I) 

We  have  also,  as  equations  of  transformation, 


(2) 


Hence  we  shall  have 

Varying  U^,  the  terms  for  the  upper  limit  become 

j  t^x"'  d   2X"   )      o.         ,     j  2X"    \     .      , 

Whence,  after  subtracting,  as  before,  F,  dx^  or  j  ^-  }  Sx^,  we 
have 

But  by  equation  (10)  of  the  last  article  these  terms  should 
become 

(2//"  -  2/'"),  Sx,  -  2/,">,"  Sx:  ;  (5) 


248  CALCULUS   OF  VARLATIONS. 

and  if  by  the  aid  of  (2)  we  express  (5)  in  terms  of  (4),  it  will 
become  identical  with  (4).  In  like  manner  we  might  treat  the 
terms  at  the  lower  limit,  only  adding  V^dx^.     We  also  have 


M 

=  2f\ 

N  = 

=  5 

d  x'" 
dy  x" 

+ 

d'  X'' 

'^  dfx" 

= 

2X'^ 

x" 

20;ir'V^' 

x^' 

•  + 

Zox'" 

x" 

=  ■ 

-  2f\ 

as  will  appear  by  consulting  the  value  of  f^  given  in  (2),  so 
that  here  also  N  —  —  M. 


Section  IX. 

DISCONTINUOUS  SOLUTIONS. 


(92.  We  now  enter  upon  a  portion  of  our  subject  which 
is  of  comparatively  recent  development,  but  is  nevertheless  of 
the  highest  analytical  importance.  But  some  general  view 
of  the  nature  of  discontinuous  solutions  having  been  presented 
in  Art.  103,  we  shall,  without  further  explanation,  proceed  at 
once  to  the  consideration  of  an  example  to  which  the  develop- 
ment of  the  subject  is  chiefly  due. 

Problem  XXXII. 

//  is  required  to  determine  the  form  of  the  surface  of  revolu- 
tion which  shall  m,eet  the  axis  of  revolution  at  two  fixed  points^ 
have  a  given  area,  and  enclose  a  maximum  solid ;  the  two  fixed 
points  being  so  taken  as  to  render  a  sphere  inadmissible. 

'  It  will  readily  appear  that  this  is  merely  Prob.  XVI.  with 
an  additional  restriction  upon  the  limits,  which  can  in  no  way 


DISCONTINUOUS   SOLUTIONS.  249 

affect  the  general  value  of  U.  .  Hence,  as  there,  we  shall,  assum- 
ing X  as  the  axis  of  revolution,  have 

and  the  limits  being  fixed,  we  must  have,  to  the  first  order, 

Now  if,  as  usual,  we  make  (J Evanish,  we  must  have 

M=2y-\-2a  Vi  +/'  -  A__^^ML^  =  O.  (2) 


To  integrate  this  equation,  write 


2ydy-\-  2a  Vi  -\- y'"^  dy  —  2ay' .  d 


yy 


Now 

f2a  VTT7'  dy  =  2a  \^VT7'y  -  f -j=y,  ^/        ' 

and 

_f2a^^_2ag^r_^l^^ 

Hence,  by  reduction,  (2)  gives 


250  CALCULUS  OF  VARIATIONS. 

and  since  the  curve  must  meet  the  axis  of  x^  c  vanishes,  and 
we  must  have 

/  H — =r^=rz  —y  \yA ^ C  =  o.  (as 

This  is  equation  (4),  Prob.  XVI. ;  and  if  we  make  dU  z^ro, 
we  are  necessarily  led  to  this  equation. 

(93.  But  the  equation  at  which  we  have  now  arrived  ad- 
mits  of  two  solutions,  j/  =  o  and  y  -\ —  =  o.    The  first, 

however,  cannot  hold  throughout  the  entire  range  of  integra- 
tion, since  the  surface  generated  is  to  be  a  given  finite  area, 
while  the  second  will,  as  we  have  already  seen,  lead  to  a 
sphere,  having  its  centre  on  the  axis  of  x^  and  is  therefore  ex- 
cluded by  the  conditions  of  the  problem. 

We  are  naturally  led  next  to  inquire  whether  the  solution 
sought  might  not  be  obtained  by  combining  in  some  manner 
the  preceding  solutions.  Thus,  in  the  figure,  let  A  and  B  be 
the  two  fixed  points.  Then  if  the  given  surface  be  less  than 
TtAB'^y  we  may  suppose  the  generating  curve  to  be  AECB\  and 
when  the  surface  exceeds  tcAB^,  we  may  suppose  the  genera- 
trix to  be  AFDB. 


Under  this  supposition  we  know  that  the  revolution  of  the 
semicircles  will  render  the  given  integral,  when  taken  from  A 
to  C,  or  from  A  to  Z>,  a  maximum,  while  the  line  CB  or  DB 
may  be  considered  as  generating  a  cylinder  whose  .diameter 
is  infinitesimal,  and  whose  surface  and  volume  are  so  likewise. 

Here,  however,  a  new  difficulty  presents  itself.     For  if  in  M 


DISCONTINUOUS  SOLUTIONS.  25  I 

we  substitute  zero  for  y,  observing  that  y  will  be  zero  also,  we 
shall  obtain  M  =  2a\  so  that  it  appears  that  y  =  o  is  not  a 
solution  of  the  equation  J/ ==  o,  and  that,  therefore,  if  this  latter 
equation  is  to  hold  throughout  the  entire  range  of  integration, 
this  solution  must  be  abandoned  also.  The  fact  is,  however, 
that  we  cannot  reject  the  solution  y  =z  o  because  it  does  not 
satisfy  the  equation  J/  =  o,  since  the  question  now  involves  a 
principle  of  variations  which  we  have  not  hitherto  considered  ; 
and  this  we  next  proceed  to  explain. 

194-.  In  former  problems  we  have  been  obliged  to  con- 
sider Sy  as  capable  of  having  either  sign,  and  therefore,  when 
6U  was  developed  into  a  series,  and  the  terms  of  the  first 
order  transformed  in  the  usual  manner,  we  were  compelled 
to  equate  M,  and  also  the  coefficients  of  Sy^,  6y^,  Sy^\  etc.,  sev- 
erally to  zero,  as  the  only  means  of  preventing  the  terms  of 
the  first  order  from  exceeding  the  sum  of  all  the  others,  and 
thus  rendering  the  sign  of  (^6^  positive  or  negative  at  pleasure. 
But  by  referring  to  Art.  99  we  see  that  the  conditions  of  this 
problem  prevent  y  from  becoming  negative,  and  hence  when 
y  is  zero — that  is,  when  the  primitive  curve  coincides  with 
the  axis  of  x — we  can  give  y  positive  increments  only. 

To  determine  in  the  most  general  manner  what  effect  this 
restriction  would  produce  when  applied  to  the  present  prob- 
lem, let  us  suppose  that  U  and  V  retain  the  same  form  as 
before,  but  that  the  limiting  values  of  x  and  y  become  vari- 
able, so  that  we  shall  have 

+J^^'Mdydx  + etc.,  (5) 

where  brackets  denote  the  entire  variation  of  U,  the  etc.  the 
terms  of  an  order  higher  than  the  first,  and  M  has  the  form 


252  CALCULUS   OF  VARIATIONS. 

given  in  (2).    If  now  we  suppose  y  to  become  zero  throughout 
the  range  of  integration,  (5)  will  become 

\PU^  =fJ\adydx-\-Qtc.=jJ'—  2r  Sy  dx -{- ttc,      (6) 


where  r  is  a  positive  constant,  since  it  appears,  by  referring  to 
Art.  99,  that  2a  must  be  essentially  negative. 

Since  the  proposed  solution  j/  =  o  does  not  reduce  the 
terms  of  the  first  order  in  [dW]  to  zero,  but  merely  to  a  single 
term,  it  is  plain  that  this  term  will  exceed  the  sum  of  all  the  fol- 
lowing terms,  and  hence  that  its  sign  will  control  that  of  \_^U\ 
But  because  dy  is  now  necessarily  positive,  the  sign  of  this 
controlling  term  is  no  longer  in  our  power,  but  is  essentially 
negative,  thus  rendering  [^W]  a  negative  quantity  of  the  first 
order.  Hence,  if  we  wish  to  render  C/a  maximum  or  a  mini- 
mum, not  as  compared  with  all  consecutive  states  of  ^  which 
can  be  produced  by  varying  y  and  y',  but  with  such  only  as 
can  be  obtained  by  making  Sy  invariably  positive  or  negative, 
we  see  that  the  solution  j/  =  o  will,  in  the  former  case,  render 
U  a  maximum,  but  in  the  latter  a  minimum.  We  may  call 
such  maxima  and  minima  conditional  maxima  or  mi7iima. 

Now  as  the  sign  of  (5^6^  will  depend  upon  that  of  the  term 
of  the  first  order,  we  have  in  this  case  nothing  to  do  with  the 
terms  of  the  second  order,  and  thus  the  problem  is  much  sim- 
plified, unless,  indeed,  these  terms  should  happen  to  become 
infinite,  which  would,  as  before,  throw  doubt  upon  the  whole 
solution.     But  this  will  not  occur  in  the  present  case. 

We  see,  then,  that  in  this  case,  by  restricting  Sy  to  one 
sign,  we  render  it  unnecessary  that  the  proposed  solution 
should  reduce  M  to  zero,  and  also  remove  the  necessity  of  an 
examination  of  the  terms  of  the  second  order.  Neither  was 
it  necessary  that  M  should  become  a  constant,  but  merely  that 
it  should  be  finite  and  of  invariable  sign.  But  should  J/ change 
its  sign,  we  could,  in  the  same  manner  as  has  already  been  ex- 


DISCONTINUOUS  SOLUTIONS.  253 

plained  for  terms  of  the  second  order,  cause  the  term  of  the 
first  order,  and  consequently  [dU\to  assume  either  sign  at 
our  pleasure. 

Simple  as  is  the  foregoing  principle  of  restricting  Sy  to  one 
sign,  it  appears  to  have  been  first  introduced  into  the  calculus 
of  variations  by  Prof.  Tod  hunter,  in  the  Philosophical  Magazine 
for  June,  1866.  It  will,  however,  when  somewhat  more  ex- 
tended, afford  the  basis  for  some  important  investigations,  and 
will  also  serve  to  explain  some  points  which  have  hitherto 
been  the  source  of  difficulty  to  the  student  in  this  department 
of  analysis. 

(95,  In  applying  this  principle  to  the  present  problem,  let 
us  first  suppose  the  given  surface  to  be  less  than  that  of  a 
sphere  having  AB  a.s  its  diameter,  and  let  the  abscissa  of  A 
be  x^,  that  of  B,  x^,  and  that  of  C,  x^.  Then  suppose  the  inte- 
gral to  be  divided  into  two  parts,  the  first  extending  from  A 
to  C,  and  the  second  from  C  to  B;  so  that  we  may  write 

^^ly^^+iy''^^-         (7) 

Now,  supposing  M  to  be  zero  throughout  the  first  integral,  its 
variation  will  reduce  to 


^a; 


and  putting  jj/  =  o,  y  =  co  ,  observing  that  2a  —  —  2r  =  —  R, 
R  being  the  radius  of  the  sphere,  this  term  will  also  vanish. 
If  we  vary  this  portion  of  the  integral  only,  while  leaving  the 
rectilinear  portion  unvaried,  we  shall,  theoretically,  be  obliged 
to  examine  the  sign  of  the  terms  of  the  second  order ;  and  we 
have  already  seen  that  this  investigation  is  not  altogether  sat- 
isfactory. Still,  as  it  is  well  known  apart  from  the  calculus  of 
variations  that  the  sphere  is  the  solid  of  maximum  volume  for 


254  CALCULUS  OF  VARIATIONS. 

a  given  surface,  we  may  assume  that  f/ will  in  this  case  become 
a  maximum ;  that  is,  that  \SU^  will  become  a  small  negative 
quantity  of  the  second  order. 

Now  throughout  the  second  integral  we  have 

M=2a=  —2r=  —  R, 

and  the  variation  of  this  integral  becomes  /     —  R6y  dx,  which 

must  be  negative,  since  Sy  is  invariably  positive ;  and  thus  in 
this  case  the  whole  integral  U  must  become  a  maximum. 

It  should  be  observed  that  while  the  values  of  x\  and  y^  are 
the  same  for  both  parts  of  the  solution,  those  of  y  for  the  same 
point  differ.  Thus  for  the  circle  y^  is  infinite,  while  for  the 
rectilinear  part  j/  is  zero,  and  we  shall  be  obliged  sometimes 
to  observe  this  and  similar  distinctions  with  great  care. 

When  the  given  surface  exceeds  that  of  a  sphere  described 
upon  AB  as  a  diameter,  let  x^  be  the  abscissa  of  D.  Then  we 
may  consider  U  as  consisting  of  two  integrals,  the  first  extend- 
ing from  A  to  D,  and  the  second  from  D  to  B,  and  we  may 
still  write 

This  mode  of  considering  the  integral  may  seem  erroneous, 
inasmuch  as  it  will  compel  us  to  regard  x  as  doubling  upon 
itself  at  D,  and  x^,  therefore,  as  greater  than  x^.  But  it  is  to 
be  observed  that  we  assume  that  U  and  SU  are  continuous 
integrals — that  is,  that  they  are  capable  of  being  expressed  by 
one  definite  integral — and  this  requires  that  x  shall  be  uninter- 
rupted. Adopting  for  the  present  this  view  of  the  subject, 
we  see,  as  before,  that  if  we  vary  the  arc  ^Z^  only,  (J  ^  must 
become  a  negative  quantity  of  the  second  order,  and  that  if 

we  vary  the  line  DB  also,  we  shall  have  6U  =  J^    —  Rdydx, 


DISCONTINUOUS  SOLUTIONS.  255 

which  is  negative,  as  before.  In  this  case  we  in  reality  reckon 
twice  the  volume  generated  by  6y  along  DB,  when  we  pass 
to  the  derived  solid. 

We  may,  however,  construct  the  two  solutions  as  in  the 
subjoined  figure,  in  which  case  we  shall  be  obliged  to  con- 
sider U  as  consisting  of  three  integrals,  and  we  have  therefore 
adopted  the  other  construction  as  being  more  easily  explained. 


B  i> 


196,  We  have  already  shown,  in  Art.  loi,  that  when  the  dis- 
continuous solution  is  necessary,  that  necessity  arises  from  the 
fact  that  the  conditions  which  require  the  surface  to  be  given, 
and  the  two  terminal  points  on  the  axis  of  x  to  be  also  assigned, 
have  been  so  fulfilled  as  to  render  them  incompatible  with  the 
general  solution.  Now  we  shall  find,  as  we  proceed,  that  dis- 
continuous solutions  generally,  if  not  always,  arise  from  some 
incompatibility  in  the  conditions  of  the  problem,  and  that  the 
conflicting*  conditions  are  imposed  sometimes  consciously, 
that  is  explicitly,  and  sometimes  unconsciously,  that  is  im- 
plicitly. The  present  problem  would,  of  itself,  afford  an 
example  of  the  former  kind,  but  it  is  in  reality  onl}^  a  comple- 
tion of  the  discussion  suggested  by  Prob.  XVI.,  and  there  the 
discontinuity  was  of  the  latter  kind,  arising  from  conditions 
incidentally  imposed,  the  effects  of  which  were  not  foreseen. 

197.  We  will  next  consider  an  example  which  will  serve 
to  extend  our  theoretical  knowledge,  and  to  prepare  us  for 
the  discussion  of  more  important  questions. 


^5^  CALCULUS  OF    VARIATIONS. 

Problem  XXXIII. 

Let  U  ^^   I     {y""^  —  2y)dx  =  /      Vdx^  and  let  it  be  required 

to  maximize  or  minimize  U,  the  limiting  values  of  x  being  fixed^ 
those  of  y  being  zero,  and  it  being  also  required  that  a  certain 
fixed  point,  whose  co-ordinates  are  x^  and  y^,  shall  not  fall  without 
the  required  curve. 

This  is,  in  fact,  merely  a  restricted  form  of  Prob.  V.,  and 
we  have 

J/—  2/^—  2.  (l) 

Now  if  it  be  possible  to  draw  between  the  fixed  points  on 
the  axis  oi  x  a  curve  satisfying  the  equation  M  =  o,  and  also 
enclosing  the  point  x^,  y^,  there  will  be  no  difficulty,  and  we 
shall  have  a  minimum  as  in  Prob.  V.  If  the  point  should 
happen  to  fall  upon  the  curve,  dy^  could  not  be  made  negative, 
but  this  would  not  affect  the  problem,  since  the  curve  would 
render  ^a  minimum  for  all  admissible  variations  oi  y  andy^ 

Suppose,  however,  that  no  curve  satisfying  the  equation 
M  =  o  can  be  drawn  so  as  to  enclose  or  pass  through  the 
point  x^,y^.  Still,  as  the  sign  of  ^y  is  wholly  unrestricted, 
except  at  the  points  x^  and  x^,  and  also  possibly  at  the  point 
x^,y^,  if  the  curve  pass  through  that  point,  we  feel  sure  that 
M  must  vanish  throughout  the  entire  integral  U.  We  are 
therefore  naturally  led  to  inquire  whether  the  solution  might 
not  be  furnished  by  drawing  from  x^  and  x^  severally  an  arc 
of  a  curve  satisfying  the  equation  M  =  o,  the  two  arcs  meet- 
ing and  not  excluding  the  point  x^,  y^ ;  and  such  a  solution 
we  now  proceed  to  consider. 

198.  Let  the  arcs  meet  at  the  point  x^,y^;  x^  and  ^fj  being 
less  than  x^.     Then  U  may  be  written 

U=£ydx+fydr,  (2) 


DISCONTINUOUS  SOLUTIONS.  257 

where  Fhas  the  same  form  as  before.  But  although  the  two 
arcs  satisfy  the  same  differential  equation  M  —  o,  still  the  con- 
stants which  enter  their  equations  cannot  be  identical ;  other- 
wise they  would  form  one  and  the  same  curve,  which  is 
contrary  to  our  present  supposition.  Hence  by  making  M 
zero  in  (i),  and  integrating,  the  general  equation  of  the  two 
arcs  may  be  written 


X 

24 


24 


(3) 


If  now  to  the  first  order  we  take  the  variation  of  each  inte- 
gral in  U  separately,  and  transform  it  in  the  usual  manner, 
observing  that  dy^  and  Sy^  vanish,  and  that  the  parts  which 
remain  under  the  sign  of  integration  must  also  vanish,  because 
M  is  zero  throughout  U,  we  shall  obtain 

SU=-  2y:"Sy,  +  2  Y:"S  Y,  +  2  Y:'S  Y:  -  2y:'Sy: 

^2y-Sy:-2Y:'SY:.  (4) 

Since  all  the  variations  in  this  equation  are  of  unrestricted 
sign,  SU  vnw^t  vanish ;  and  also  if  (4)  be  expressed  so  as  to  in- 
volve only  variations  which  are  entirely  independent,  the  co- 
efficients of  these  variations  must  severally  vanish.  Now  if 
we  assume  that  x^  does  not  vary,  Sy^  and  dY^  are  the  same 
quantity.     Moreover,  from  (3),  we  have 

y:":^x,^ec,         and     Y:"  =  x,^eg.  (5) 

Whence 

-2y:"Sy,-\-2Y:"dY,=  i2{g-c)Sy^',  (6) 

and  since  Sy^  is  certainly  an  independent  variation,  c  and  g 
must  be  equal. 


25 o  CALCULUS  OF  VARLATLONS. 

Now  consider  the  t^xvcis  2y^'Sy^  —  2Y^'SY^.  To  make 
these  terms  vanish  we  may  have  jg^'  =  Y^' ,  and  also  ^y^—  ^  Y^\ 
or  Iz^  =  o  and  Y/^  =  o.  Under  the  first  supposition  y^^  and  Fg' 
must  mean  the  same  thing,  otherwise  their  variations  would 
not  be  necessarily  equal. 

If  now  we  equate  jj/g'^  to  Y^'\y/  to  Y/,  and/s  to  Fg,  taking 
the  values  of  these  quantities  found  by  differentiating  (3),  we 
easily  discover  that  if  c  and  £•  are  equal,  c^  =  g-^,  c\  =  g^  and 
^3  =  ^3>  which  is,  as  has  been  shown,  not  admissible  in  this  case. 

But  suppose  jg^^  =  o  and  Fg^^  =  o.  Then  y^  and  Fg^  need  not 
mean  the  same  thing,  and  we  have  only  c  ^^  g  and  c^  ■=  g^.  But 
there  still  remain  in  (4)  the  terms  2Y^'SYI  —  2y^" Sy^\  and  to 
make  these  vanish  we  must  have  Y^"  =  o  and  yj'  =  o.  Take 
the  origin  midway  between  the  points  x^  and  :i\,  and  let  x^  =  e 
and  ;t:o  =  —  ^.  Equating  the  values  of  y^^'  and  Y/\  as  found 
from  (3),  we  have 

-  +  ^£-^+2g  =  --6ce  +  2c,. 

Whence,  since  c  ^  g  and  c^  =  g^,  c  and  g  are  zero. 

The  equation  Y^'  =  o  now  becomes  — h  2^1  =  o,  and  the 

2 

equation  Fg^^  —  o  gives  —^-\-2g^=z  o,  impossible  equations  un- 
less e""  and  x^^  be  equal,  which  they  cannot  be  since  x^  was 
taken  numerically  less  than  x^.  Hence  we  must  abandon  this 
solution,  since  it  will  neither  cause  SW  to  the  first  order  to 
vanish  nor  to  have  an  invariable  sign. 

(99.  There  remains  but  one  supposition,  which  is  that  the 
arcs  be  drawn  as  before,  but  meet  at  the  point  x^,  y^ ;  and 
this,  which  we  shall  find  to  be  the  correct  solution,  we  next 
proceed  to  consider. 

It  is  plain  that  C/and  also  ^C/ will  have  the  same  form  as 
before,  except  that  the  suffix  3  w^ill  be  changed  into  2.    But  it 


DISCONTINUOUS  SOLUTIONS  259 

will  not  now  be  necessary  to  make  all  the  terms  in  ^ Evanish, 
because  6y^  and  S  K,  are  the  same  quantity,  and  dy^  is  neces- 
sarily positive.  Now,  as  before,  the  terms  involving  these 
two  variations  will  become  12  {g—  c)  Sj/^,  and  it  will  be  neces- 
sary to  make  the  remainder  of  (5^  C/ vanish,  because  it  contains 
only  variations  of  unrestricted  sign. 

We  have  then  to  examine  whether  we  can  make  these 
terms  vanish  ;  and  if  so,  what  will  then  be  the  sign  oi  g  —  c. 

To  make  the  terms  involving  6y^'  and  d  F/  vanish,  we  have, 
as  before,  . 

F/'=j/'        and     F;=j/,'.  (7) 

The  first  equation  being  necessary,  and  the  second  also  neces- 
sary unless  F/^  =  o  and  j//'  =  o,  a  case  which  we  shall  subse- 
quently consider. 

To  make  the  terms  involving  d  F/  and  Sy^'  vanish,  we  have, 
as  before, 

F/^  =  o        and    y/'  =  o.  (8) 

We  have  also,  from  the  other  conditions  of  the  question, 

7o  =  o,         Y,  =  o,         V,=f,;  (9) 

and  these  equations,  together  with  (3),  which  still  holds,  will 
be  sufficient  for  our  purpose. 

200.  Take  the  origin  as  before,  and  also  denote  x^  by  a, 
and  jTg  by  d.  Then  finding,  from  (3),  the  successive  differen- 
tials of  y  and  F,  (7),  (8),  and  (9)  give,  by  substitution. 


—  +  6ac-\-  2c,  =  --  +  6^^+  2g,, 


(10) 


26o 


CALCULUS  OF  VARIATIONS. 


3  3 

^+6^^  +  2^x=0,         ^--6ce^2c,  =  o\  (II) 


24 


24 


^^^  -j-  iTj^''^  —  ^2^  -|-  ^3  =  o ; 


(12) 


24 


24 


+^'«'+<ri^'+<r3^ +^3  =  ^. 


+    ^<^^  -f"    ^l^'^  +  ^3<^    +    ^3  =    ^. 


Now  from  (11)  we  have 


^1  =  -  3<^^ 


and     c^  =  ye , 


and  the  first  of  equations  (10)  becomes 


(13) 


(14) 


(15) 


and  solving  for  g-e,  and  then  adding  g-e  to  both  members,  we 
can  obtain 

_    2eg- 


g-c 


a-\-  e 


(16) 


But  e  is  positive ;  and  estimating  x  toward  a,  a  will  also  be 
positive,  so  that  the  sign  of  ^  —  <:  will  be  the  same  as  that  of 
g,  which  we  must  next  determine. 

Subtracting  (12)  from  (13)  and  dividing  hj  a  —  e,  we  have 


,^3  +<^i(^  +  ^)  +<^(^'  +  ^^  +  e") 


g^^g{a'-2ae-2e')-\- 


a'  +  a'e-^ae'  +  e' 


24  a  —  e 


24 


(17) 


DISCONTINUOUS  SOLUTIONS.  26 1 

the  last  member  being  obtained  by  substituting  for  g^  its  value 
from  (14).     In  like  manner,  from  (12)  and  (13),  we  obtain 

c,-\- c(a' -^  2ae  -  2e')A^ -^ XA_  ^  ^^.    (18) 

Now  subtracting  (18)  from  (17),  substituting  for  c  its  value 

^ — ; — ^-  from  (15),  and  reducing  the  second  member  to  a  com- 

a  A^  e 

mon  denominator,  we  have 


—  c^-^ g  \  a^  —  2ae  —  2e^  —  {a"  +  2ae  —  2/)  \ 

t  a  -\-  e  ) 

+ 


+ 

a^e  —  5^^  _     2be 
12       ~  a"  — / 


From  (14)  and  (15)  we  have 

c.-g.=  2>e{c-\-g\  (20) 

Now  from  the  second  of  equations  (10),  we  obtain 
g,-c,  =  2a{c,  -  g)  +  ia\c  -  g) 

=  6ae{g+  c)  -  3a\g-  c)  =  |^,  (22) 

the  second  member  being  obtained  by  (20),  and  the  third  by 
(21)  and  (16).     Hence,  by  substitution,  (19)  becomes 

a'e  —  ^'   ,   '^V  —  5^'  _     2be 


262  CALCULUS  OF  VARIATIONS. 

Whence 

Now  we  will  so  estimate  y  as  to  make  b  positive.    Then,  since 
e  exceeds  a,  to  make  g  positive,  we  must  have 


or 


2-^x5^'- 

'        12 

24 

4         24* 

(23) 


But  in  Art.  42  we  showed  that  the  general  equation  of  the 
single  curve  meeting  x  at  the  points  —  e  and  +  ^j  J'/  and  j^' 
not  being  fixed,  is 

-^""24        4~       24* 

Hence,  if  the  members  of  (23)  were  equal,  b,  or  y^,  would  be  an 
ordinate  of  a  single  curve  drawn  from  the  pomt  x^  to  ;r,,  and 
satisfying  all  the  other  conditions  of  the  problem.  But  since 
y^  or  b  is  m  this  case  too  great  to  be  made  the  ordinate  of  any 
such  single  curve,  the  conditions  of  (23)  are  fulfilled,  and  g 
must  be  positive. 

Therefore,  in  this  case,  ^C/ reduces  to  \2{g—  c)dy^,  which, 
because  Sy^  cannot  become  negative,  is  positive,  so  that  these 
arcs  furnish  the  minimum  solution  required. 

201.  We  have  still  to  consider  the  casein  which j/'  =  o 
and  IV'  =  o.  Here  the  second  of  equations  (10)  is  not  neces- 
sarily true,  but  the  first  may  be  replaced  by  the  two  equations 

-  -\-  6ag  -\-  2g  =  o        and     -  +  6ac  -\-  2c^  =  o.         (24) 
2  2 


DISCONTINUOUS  SOLUTIONS.  263 

Then  from  (11)  and  (24)  we  readily  deduce 

a-\-  e  J  •  a  —  e 

p-  = ' —  and     r  = . 

^  12  12 

Hence  d^^  becomes  in  this  case 

dU^  I2{g—  c)  dy^  ^  —  2e  (5>,, 

which,  as  ^y^  is  still  positive,  must  be  negative. 

Thus  it  appears  that  this  solution  will  render  U  a  maxi- 
mum. Now  in  the  present  case  g  is  necessarily  negative,  and 
we  have  seen  that  when  Y^'  =  y^'  and  F/  =  7/,  g  is  neces- 
sarily positive.  Hence,  when  we  satisfy  the  first  of  these 
equations  by  making  Y^'  and  y^'  severally  vanish,  the  second 
cannot  hold  true.  We  see,  then,  that  the  minimum  solution 
consists  of  two  arcs  which  satisfy  the  equation  M  =  o  and 
meet  at  the  point  x^,  y^,  so  as  to  have  there  no  abrupt  change 
of  direction,  and  to  make  their  radii  of  curvature  at  that  point 
equal  and  finite. 

202.  In  closing  this  discussion  we  must  observe,  first,  that 
when  we  propose  to  make  the  two  arcs  meet  at  the  point 
;r„,  y^,  it  is  by  no  means  the  same  as  if  we  had  been  required 
to  draw  each  arc  so  as  always  to  pass  through  the  two  fixed 
points.  For  then  y^  would  have  no  variation,  and  we  would 
treat  each  curve  separately  by  the  well-known  rules  of  varia- 
tions. 

In  the  second  place,  the  terms  maxima  and  minima  are 
here  also  used  in  the  technical  sense  already  explained,  and 
we  must  be  careful  not  to  say  that  the  present  solution  gives 
the  least  value  of  W.  For  B  being  the  point  x^,  y.^,  we  can,  by 
a  construction  hke  that  of  the  figure,  make  U  as  small  as  we 
please. 


264  CALCULUS   OF  VARLATIONS. 

All  that  follows  from  the  preceding  discussion  is,  that  if 
we  draw  two  arcs  as  required  by  the  solution,  and  then, 
regarding  this  curved  line  as  a  primitive,  pass  to  any  other 
curved  line  which  can  be  derived  from  the  first  by  infinitesi- 
mal variations  of  y  and  y" ,  the  variation  of  y^  being  positive, 
U  will  be  thereby  increased  by  a  quantity  of  the  first  order. 
If  we  make  dy^  zero,  the  proposed  solution  will  reduce  the 
terms  of  the  first  order  to  zero,  and  we  shall  be  obliged  as 
usual  to  appeal  to  those  of  the  second  order,  which  will  be 


dU=  r^dy"^  dx  +  r^d  Y"''  dx, 


which,  being  positive,  will  render  U  in  this  case  also  a  mini- 
mum. , 

203.  We  may  now  with  profit  consider  partially  the  gen- 
eral theory  of  discontinuous  solutions. 

Suppose  we  wish  to  determine  the  relations  between  x 
and  y   which    will    maximize    or    minimize    the    expression 

U  —  J^     Vdx,  V  being,  as  usual,  any  function  of  x,  y,  y',  etc. 
Then,  after  the  usual  transformation,  we  may  write 


SU=L,—  L,-\r  r^M dy dx, 


where  L,  and  L^  have  the  well-known  form  of  the  terms  at  the 
limits.  Now  if  no  restriction  be  imposed  upon  ^y,  we  know 
that  M  must  vanish  throughout  the  entire  range  of  integra- 
tion, and  likewise  Z^  and  L^  must  vanish. 

But  suppose  the  problem  be  such  that  ^y  must  always  be 
positive  or  always  negative ;  then  it  may  not  be  necessary  to 
make  M  vanish,  provided  it  be  of  invariable  sign,  and  pro- 
vided, also,  that  the  terms  at  the  limits  either  vanish  or  be- 
come of  the  same  sign  as  the  unintegrated  part;  in  which  case 


DISCONTINUOUS  SOLUTIONS.  265 

(J  C/ will  become  a  quantity  of  the  first  order,  and  there  will 
be  no  need  of  examining  the  terms  of  the  second  order. 

Suppose,  next,  that  U  is  such  that  it  may  naturally  be 
divided  into  a  number  of  integrals,  say  n,  the  first  extending 
from  x^  to  x^,  the  second  from  x^  to  x^,  etc.,  the  last  extending 
from  x^n_^  to  x^ ;  and  suppose  dy  is  of  invariable  sign  through- 
out one  or  more  of  the  intervals  into  which  x  is  divided,  but 
is  unrestricted  throughout  the  others.  Then  M  must  vanish 
throughout  the  latter ;  but  if  throughout  each  of  the  former 
M  be  of  invariable  sign,  and  if  the  sign  of  Mdy  be  the  same 
throughout  each,  M  need  not  vanish  provided  certain  con- 
ditions can  be  secured  at  the  limits,  and  we  shall  have  a  dis- 
continuous solution,  made  up  of  curves  satisfying  different 
differential  equations. 

But  when  the  sign  of  ^y  becomes  necessarily  invariable 
throughout  any  interval,  we  shall  find  that  this  restriction 
results  from  the  fact  that  there  is  throughout  that  interval 
some  boundary  which  the  required  curve  is  forbidden  to  pass ; 
and  in  order  that  the  sign  of  Sy  may  be  made  invariable  by 
this  boundary,  the  required  curve  must,  throughout  that  inter- 
val, coincide  with  it.  It  will,  therefore,  readily  appear  that 
whenever  any  portion  of  the  required  solution  does  not  satisfy 
the  equation  M  —  o/\t  can  consist  of  nothing  but  a  portion  of 
some  boundary,  the  nature  of  which  will  be  generally  known. 
Thus  in  the  case  of  a  sphere,  this  boundary,  although  not  ex- 
plicitly assigned,  is  easily  seen  to  be  the  axis  of  x,  the  implicit 
condition  that  y  is  not  to  become  negative  making  this  the 
boundary  below  which  y  cannot  pass. 

If,  however,  the  sign  of  ^y  be  restricted  for  some  point  or 
points  only,  as  in  the  preceding  problem,  the  equation  M  =  o 
must  hold  throughout  U,  although  the  equations  of  the  arcs 
for  different  intervals  may  differ  widely  in  the  values  of  the 
constants  which  they  contain. 

It  will,  we  think,  now  be  evident  that,  in  general,  when  a 
discontinuous  solution  presents  itself,  it  will  be  made   up  in 


266  CALCULUS  OF  VARLATLONS. 

one  of  these  three  ways :  first,  some  combination  of  arcs  satis- 
fying the  equation  J/  =  o ;  second,  some  boundary  or  certain 
boundaries  ;  third,  some  combination  of  this  boundary  or  these 
boundaries,  with  arcs  satisfying  the  equation  M  =^  o. 

204.  Let  us  now  consider  the  integrated  part  of  dU,  when 
U  is  divided  as  explained  above. 

As  the  different  portions  of  the  discontinuous  solution  meet 
at  the  points  whose  abscissae  are  x^,  x^,  etc.,  jk  will  have  the 
same  value  for  two  curves  meeting  at  those  points,  but  the 
values  of  y ,  y^,  etc.,  for  two  curves  at  their  points  of  meeting 
may  differ  widely.  To  recognize  this  distinction,  we  employ 
the  suffixes  2  and  3  to  denote  quantities  both  of  which  corre- 
spond to  x^,  but  belong  to  different  curves  meeting  at  the 
point  x^,y^,  and  we  divide  x  into  x^,  x^,  x^,  etc.,  the  last  being  x^, 
the  suffixes  3,  5,  etc.,  being  reserved  for  the  second  of  the  two 
quantities  corresponding  to  x^,  x^,  etc. 

Now  performing  the  integration  for  each  integral  sepa- 
rately, the  first  gives  L^  —  Z^,  the  second  L^  —  Zg,  etc.,  so  that 
the  entire  integrated  part  of  (^^  becomes 

Zj  —  Zo  +  Z.  —  Zg  +  Z,  — Z54- etc. +Z2n_2—Z2^_„  or  Z.    (i) 

Now  if  all  the  variations  involved  in  Z  be  of  unrestricted 
sign,  it  must  vanish ;  and  also  if  Z  be  transformed  so  as  to  con- 
tain independent  variations  only,  the  coefficients  of  these  vari- 
ations must  severally  vanish.  But  suppose  some  of  the  varia- 
tions involved  in  Z  to  be  of  restricted  sign.  Then,  the  other 
terms  having  vanished  as  before,  it  may  not  be  necessary  to 
make  these  terms  vanish  also.  For  if  these  restricted  varia- 
tions be  related,  suppose  them  to  have  been  reduced  to  inde- 
pendent variations,  and  let  H,  Z,  K^  etc.,  be  the  several  pro- 
ducts of  each  variation  and  its  coefficient.  Then  we  can 
reduce  any  of  the  quantities  H,  Z,  etc.,  to  zero  by  making  its 
variation  factor  vanish.  If,  therefore,  these  quantities  be  all  of 
Uke  sign,  that  of  Z  is  determined ;  but  if,  on  the  contrary, 
they  be  not,  Z  can  be  made  positive  or  negative  according  as 


DISCONTINUOUS  SOLUTIONS.  267 

we  reduce  to  zero  the  negative  or  positive  quantities.  But 
suppose  i/,  /,  etc.,  to  be  of  like  sign,  making  that  of  L  the 
same,  and  that  this  sign  does  not  conflict  with  that  of  M 6y  in 
the  unintegrated  part.  Then-  dU  becomes  a  quantity  of  the 
first  order,  having  a  fixed  sign,  and  we  need  not  examine  the 
terms  of  the  second  order.  But  if  H,  /,  etc.,  be  of  unHke  sign, 
or  if  their  sign,  when  the  same,  conflict  with  that  of  Mdy,  L 
must  vanish  altogether. 

In  equation  (i)  we  have  assumed  that  no  portion  of  the 
axis  of  X  is  to  be  counted  twice,  as  in  Prob.  XXXI I.,  where 
the  sphere  extends  beyond  x^,  because  such  cases  will  seldom 
occur.  When,  however,  they  do  arise,  Z,  although  differing 
somewhat  in  form  from  (i),  can  be  readily  found  by  integrat- 
ing each  portion  separately,  as  before ;  and  then  all  the  condi- 
tions which  we  have  just  explained  will  hold  true  for  this  case 
also. 

Problem  XXXIV. 

205,  It  is  required  to  determine  what  will  be  the  solution  of 
Prob.  VI I.  when  the  two  fixed  points  are  so  taken  that  no  catenary 
ca7i  be  drawn  between  them  having  its  directrix  on  the  axis  of  x. 


Here 


U 


lyv.^-y'^dx^Sydx        (I) 


and 


Now  it  is  natural  to  inquire,  first,  whether  any  restrictions 
have,  either  expUcitly-or  implicitly,  been  imposed  upon  the 
sign  of  Sy^  in  virtue  of  which  the  equation  J/ =  o  need  not  hold 
throughout  U.  For  if  not,  the  solution  can  consist  of  noth- 
ing that  will  not  satisfy  this  equation.     Now  Vdx  in  (i)  is  the 


268  CALCULUS  OF  VARIATLONS. 

value  of  any  element  of  the  generated  surface  divided  by  27r, 
and  it  does  not  seem  reasonable  to  suppose  that  this  surface 
can  ever  become  negative.  Hence,  since  we  take  Vi-^-y'"^ 
positively,  it  would  appear  that  y  cannot  be  negative ;  that  is, 
that  dy,  along  the  axis  of  x,  must  be  positive. 

We  infer,  then,  that  should  the  solution  contain  anything 
which  does  not  satisfy  the  equation  M  —  o,  it  can  only  be 
some  portion  of  the  axis  of  x,  and  that  such  portions  will  be 
likely  to  occur. 

206.  Let  us  next  examine  the  equation  i)f=  o  to  see  what 
can  be  obtained  from  this  source. 
This  equation  will  give 


/  vY+y^  dy  =  vr+y-^y  -f-y^.  ^/ 


^'''+y'-':7=k^^  +  fy^-: 


yy 


Vi-\-y"  y       Vi+y 

Whence 


c. 


y      _ 


Vi  +y' 


(3) 


This  is  the  same  as  equation  (2),  Art.  59,  so  that  this  is  the 
only  condition  which  can  be  obtained  from  the  equation 
M=o. 

Suppose  now  that  we  digress  from  the  method  of  solution 
pursued  in  Art.  59,  and  make  c  zero.  Then  (3)  will  give  either 
jj/  —  o  or  y  =00  ,  and  these  two  solutions,  although  neither  can 
be  employed  alone,  can  be  combined..  For  let  A  and  B  of 
the  figure  be  the  two  fixed  points,  CD  being  the  axis  of  x. 
Then  the  discontinuous  solution  proposed  will  be  the  broken 
line  A  CDB. 


DISCONTINUOUS  SOLUTIONS.  269 

Thus  in  this  case,  as  in  Prob.  XXXI I.,  the  solution  y  =  o, 
which  arises  from  the  same  conditions  in  both,  is  suggested  as 
one  solution  of  the  equation  J/  =  o,  which  it  does  not,  how- 
ever, in  either  case  satisfy.      But  this  suggestion  was  not 


necessary,  as  this  solution  was  anticipated  by  the  reasoning 
of  the  preceding  article,  which  would  be  equally  applicable 
to  Prob.  XXXII. 

207.  We  now  proceed  to  show  that  the  proposed  solution 
will  minimize  U.  As  we  cannot  treat  infinite  quantities  by 
the  methods  of  variations,  we  shall,  to  avoid  their  occurrence, 
transform  to  polar  co-ordinates.  Take  some  point  within  the 
figure  as  the  pole,  the  initial  line  being  parallel  to  CD.  Let 
V,  the  angle  between  r,  the  radius  vector,  and  this  initial,  be 
estimated  in  the  direction  A  CDB,  and  let  k  be  the  distance 
of  the  pole  from  CD.  Then  any  element  of  the  generating 
curve  will  be  Vr"  ^r'"^  dv,  and  its  distance  from  CD  will  be 
k  —r  sin  V.     Then,  ds  being  an  element  of  the  surface. 


ds  =  27t{k  —  r  sin  v)  Vr^  -\-  ^  dv. 
Whence 

U=S^yk-rsxnv)^^^^^^dv=£ydv  (4) 

and 


=  X''  I  ~"  ^^'  +  ^''  sin  vSr-^zrdr-^  zr'Sr'  [  dv.  (5) 


270  CALCULUS   OF  VARIATIONS. 

But  since  the  proposed  solution  cannot  be  represented  by  the 
same  equation  throughout,  (3)  and  (4)  must,  without  in  any 
manner  changing  their  form,  be  written  as  three  integrals — 
that  is,  three  times  with  different  limits — the  first  portion,  A  C, 
extending  from  v^  to  v^ ;  the  second,  CD,  from  v^  to  v^ ;  and  the 
third,  DB,  from  v^  to  v^.  Then,  transforming  6U  in  the  usual 
manner,  we  have 

6U=  {zr'Sr)-  {zr'Sr),-^  {zr'Sr\-  {zr'dr\-\-  {zr'dr),-  {sr'dr\ 

-f  fj^'  I  _  Vr'  +  r''  sinv  +  zr  —  -^  zr'  \  drdv.  (6) 

Now  the  suffixes  2  and  3  relate  to  C  as  being  on  the  two 
lines  AC znd  CD,  and  the  same  is  true  of  the  suffixes  4  and  5, 
so  that  r,  =  r,,  v^  =  v„  r,  =  r„  v,  =  v„  Sr,  =  dr„  Sr,  =  Sr, ; 
while  r/  and  r^'  differ,  as  do  also  r/  and  r/.  Now  dr^  and  6r^ 
are  zero,  the  points  A  and  B  being  fixed ;  and  although  the 
other  suffixed  variations  need  not  vanish,  still  at  the  points  C 
and  D,  for  either  line,  we  have  k  —  r  sin  v  =  o\  so  that  all  the 
integrated  terms  in  (6)  disappear,  and  we  have  left  only  the 
integral,  which  must  be  considered  as  divided  into  three  parts, 
as  just  explained. 

208.  We  may  now  write 

d  U  =fJ^MSr  dv  -^fJ'Mdr  dv  -{-fJ^MSr  dv,  (7) 

where 


M=.  —  Vr'  +  r''  sin  V  +  zr  -  -^  zr\  (8) 

dv  ^  ^ 

Now  consider  first  the  second  integral.  Along  CD  we  have 
y^  —  r  sin  ^  =  o,  so  that  M  —  —  X^r"  +  r""  sin  v,  a  negative 
quantity  of  invariable  sign.     But  along  this  Hue  dr  is  always 


DISCONTINUOUS  SOLUTIONS.  2/1 

negative,  so  that  every  element  of  this  integral  becomes  a 
small  positive  quantity  of  the  first  order. 

Let  us  now  examine  the  sign  of  the  first  integral.     Along 
AC  WG  have  r  cos  v  a  constant,  say  c,  so  that  we  find 

r  sin  V       c  sin  v 


cos  V         cos'^  V  *  COS^  V 


zr  = 


r  = = ^-,  Vr  4-  r '  ^ 

cos  Z^  COS   V 

kr  cos^  7/       r^  sin  z^  cos^  ^ 


c  c  r  cos  z/ 

=  k  cos  V  —  r  cos  27  sin  v  =^  k  cos  v  —  c  sin  ^'. 
Whence 


J/  = ^—  4-  /^  cos  V  —  c  sin  ^' 

cos  z^    ' 


dv 


. ,  •      \  sin  e; ) 

\k  cos  V  —  c  sm  z') y 

cos  2^  ) 


c  sin  ^'  •         ,    7  </  (  ,    .  c  sin  27 ) 

= c  sin  V  -\-  k  cos  2^  —  — —  ^  ye  sin  z/ y  . 

cos  V  dv  {  cos  V   \ 

Differentiating  the  first  term  within  the  parenthesis,  and  also 
putting  for  sin^  v,  i  —  cos^  v,  we  may  write 


,^           ^  sin  ^  .         X     d   \ 

M  = ^  sm  ^  4-  ^-  A 

cos  27 


.     d  \      c  \ 

-{--T-\ ^  cos^  y , 

dv  [  cos  V  ) 


which  will  be  found  by  differentiation  to  reduce  to  zero. 
Similarly  we  shall  find  that  M  will  vanish  along  the  line  BD ; 
so  that  if  we  vary  the  whole  line  ACDB,  or  CD  only,  SU  will 
become  a  positive  quantity  of  the  first  order,  and  we  have  a 
minimum. 

209.  If,  however,  the  fine  CD  be  not  varied,  we  cannot, 
since  the  terms  of  the  first  order  vanish  along  AC  and  BD, 


2/2 


CALCULUS  OF  VARLATLONS. 


assert  that  U  will  be  a  minimum  without  examining  the  terms 
of  the  second  order;  and  this  we 'next  proceed  to  do. 

Putting  u  for  ^r"  +  r''  and  Z  for  k  —  r  sin  v,  these  terms 
are 


) 

[  L 


2r  sin  V   ,    Zr'^' 


Sr' 


+D 


2r'  sin  V       2Zrr'' 


2«^'^o 


(9) 


Now  the  second   integral  in  (9)  can  never  be  negative,  so  that 
we  need  only  transform  the  first.     We  have 


rr'  sin  V  .     0.  /  /    _  ^r'r'  sin  v        f*  ^     ^^  '^'  ^in  v  Sr 

^         u  '  u  ^        dv  2L         ' 


dv 


Sr^r'  sin  v 


.  f  ^.J^  Sr  Sr'dv  -  fsr'  #  ^l^Hi^.  dv. 


^  dr  Sr'dv  = 


Sr^r'  sin  v 


f 


Sr' 


d  r'  sin  V 
dv       It 


.dv.  (10) 


Now  we  must    observe  that  each  integral  is  to  be  consid- 
ered as  divided  into  three  integrals,  identical  in  form  but  with 

different  limits,  so  that  the  term  Sr  will,  as  usual,  ap- 

pear  with  the  sufhxes  o,  i,  2,  3,  4  and  5.     But  since  Sr  now 
becomes  zero  at  the  points  C  and  D,  as  well  as  at  A  and  B, 


DISCONTINUOUS  SOLUTIONS.  273 

all  these  limiting  terms  must  vanish,  and  then  by  the  aid  of 
(10)  the  first  integral  in  (9)  becomes 

_  p  (  r^._  _l_d_ r^nj;  |  ^^^^  ^  _  T'^^^^^,  („) 

«^^-o       [  U  2  dv  U  )  «^^o 

Now  along  A  C  we  have,  as  before, 

c  ,       c  sin  V  ^  /     \ 

rco^v  —  c,         r= ,         r— — -,         u  = --,       (12) 

cos  V  cos  V  cos  V 

so  that  along  this  line  we  have 

N  =  sin  z>  cos  v cos'  v  =  o, 

2  dv 

and  similarly,  N  will  also  vanish  along  DB^  because  there  we 
shall  have  r  cos  v  ■=  —  c.  Thus,  finally,  since  ^r  and  ^r'  are 
zero  along  CD,  the  terms  of  the  second  order  reduce  to  the 
second  integral  in  (9),  which,  as  we  have  already  seen,  can 
never  become  negative. 

210.  It  is  plain  that  the  discontinuous  solution  which  we 
have  just  examined  exists  even  when  the  fixed  points  are  so 
taken  that  a  catenary  is  admissible.  But  to  determine  in  this 
case  which  of  the  minima  gives  to  U  the  smaller  value  is  a 
problem  of  the  differential  and  integral  calculus  solely,  and  for 
this  purpose  we  have  the  following  formulse.  For  the  discon- 
tinuous, 

s  being  the  entire  surface.  For  the  continuous,  let  PT  be  any 
line  tangent  to  the  catenary  at  P,  and  meeting  the  axis  of  x  at 
T,  the  abscissa  of  which  is  x^,  and  let  5  denote  the  surface 
generated  by  PT,  while  s  denotes  that  generated  by  the  por- 
tion of  the  catenary  between  P  and  its  lowest  point.     Then, 


274  CALCULUS  OF  VA^UATLONS. 

regarding  x^  as  positive  or  negative  according  as  P  and  T  are 
on  the  same  or  opposite  sides  of  the  axis  of  y.,  when  it  passes 
through  the  lowest  point,  we  have 

in  which  a  is  the  well-known  constant  of  the  equation  of  the 
catenary,  and  can  be  calculated  approximately  when  the  co- 
ordinates of  the  fixed  points  are  given.  It  will  be  found,  how- 
ever, that  sometimes  the  continuous,  and  sometimes  the  dis- 
continuous solution  will  generate  the  smaller  surface.  ^ 

211.  We  have  seen  by  the  reasoning  of  Art.  5  that  the  cal- 
culus of  variations  is  not  theoretically  bound  to  furnish  all 
possible  solutions ;  and  since  two  may  exist  in  the  present 
problem,  it  is  natural  to  inquire  whether  there  may  not  be 
another,  which  will  render  the  surface  less  than  does  either 
of  those  which  we  have  considered.  We  reply  that,  while 
theoretically  such  might  be  the  case,  still  no  such  solution  has 
ever  been  discovered,  and  there  would  seem  to  be  little  doubt 
that  one  of  the  two  already  examined  will  always  give  the 
least,  as  well  as  a  minimum  value  of  U. 

In  fact,  we  are  now  beginning,  and  shall  continue,  to  verify 
the  remarks  of  Art.  14,  and  to  show  that,  although  subject  to 
some  restrictions  which  would  seem  to  greatly  limit  its  power, 
the  calculus  of  variations  is  in  reality  capable  of  furnishing 
nearly  all  the  solutions  pertaining  to  the  maxima  and  minima 
states  of  irreducible  integrals.  We  shall  find,  moreover,  that 
these  solutions  will  generally  in  some  way  present  themselves 
as  solutions  of  the  equation  M—o,  although  they  may  in 
reality  not  satisfy  that  equation  at  all. 


DISCONTINUOUS  SOLUTIONS.  2/5 


Problem  XXXV. 


212.  A  projectile  which  is  acted  upon  by  gravity  alone  is  to 
start  from  one  fixed  point  and  to  pass  through  another.  It  is  re- 
quired to  determine  the  nature  of  its  path,  so  that  the  action  7nay 
be  a  minimum.'^ 

Assume  the  origin  at  the  starting-point,  and  estimate  x 
vertically  downward,  and  let  the  initial  velocity  be  s/2ga. 
Then  we  know  that  the  velocity  of  the  projected  particle  at 
any  point  of  its  path  will  be  ^2ga(xT\-  a).     Hence 

u  =X"  n^ +«)(!+/')  dx  =£'  vdx.        (I) 

Whence,  in  the  usual  way,  we  obtain 

So  that 

X  ^  a  —  c 
and,  by  integration, 


y  —  c^—  ±2  Vc{x  -\-a  —  c).  (3) 

Since  X  and  y  are  simultaneously  zero  at  the  starting-point,  we 
have 


—  c^=  ±2  Vc{a  —  c) 
and  (3)  becomes 


y±2  Vc{a  —  c)  =  ±2  Vc(x+  a  —  c).  (4) 


*  See  Todhunter's  Researches,  Chap.  VIII. 


2/6  CALCULUS   OF  VARLATLONS. 

If  c(a  —  c)  be  positive,  (4)  represents  two  parabolas,  and 
we  must  now  consider  whether  these  parabolas  can  be  made 
to  pass  through  the  second  point.     We  have 


y,±2  Vc{a  -c)=  ±2  \/c{x,  +  a  —c\  (5) 

so  that,  squaring,  we  obtain 


Whence 

( J,'  -  ¥x^"  =  i6y,'c(a  -  c). 

Hence  it  appears  that  c{a  —  c)  can  never  become  negative,  and 
(5)  will  therefore  contain  no  imaginary  quantity,  unless  c  be- 
come imaginary.     But  the  last  equation  may  be  written  thus: 

i6c\x:  +  j^)  -  ^cy^ix,  +  2a)  +y:  =  o ; 

or,  dividing  by  the  coefficient  of  ^^  i^  may  be  written 

c'-2Pc+Q^o.  (6) 

From  (6)  we  may  obtain  the  values  of  c,  which,  since  P 
and  Q  are  positive,  will  be  real  so  long  as  Q  does  not  exceed 
P^  the  two  roots  being  equal  when  P'  =  Q.  Now  the  condi- 
tion P'  >  or  =  Q  gives,  by  reduction, 

y,' <     or     =4a{x,-\-a).  (7) 

Hence  we  see  that  if  the  first  member  of  (7)  exceed  the  sec- 
ond, c  in  (5)  can  have  no  real  value ;  if  the  members  become 
equal,  c  can  have  but  one  value ;  and  if  the  first  member  be- 
come less  than  the  second,  c  can  have  two  real  values. 

Now  it  is  evident  that  for  any  given  values  of  x^,  y^  and  a, 
but  one  of  the  forms  of  (5)  can  be  true  for  the  same  value  of 
c,  and  that  therefore  we  can  have,  passing  through  the  two 


DISCONTINUOUS  SOIUTIONS.  2jy 

fixed  points,  as  many  parabolas  as  there  are  real  values  for  c. 
But  (7),  when  its  members  are  equal,  is  itself  the  equation  of 
a  parabola,  which  may  be  called  the  limiting  parabola.  For 
we  see  that  if  the  second  point  lie  without  this  parabola,  it 
cannot  be  joined  to  the  first  by  any  parabola  which  will  sat- 
isfy all  the  conditions  of  the  question,  so  that  the  solution,  if 
one  exists,  must  be  discontinuous.  If  the  point  be  on  this 
parabola,  one  parabola  only  can  be  drawn ;  w^hile  if  the  point 
be  within  the  limiting  parabola,  two  parabolas  can  be  drawn. 
Of  course  when  the  values  of  ^,,  j,  and  a  are  fully  given, 
we  can  determine  the  one  or  two  equations  involved  in  (5),  so 
that  they  may  be  without  ambiguity  of  sign.  But  when  two 
values  of  c  exist,  we  cannot  determine  which  must  be  taken, 
unless  we  fix  the  angle  which  the  projectile  in  starting  makes 
with  the  horizontal,  two  angles  being  admissible. 

213.  Let  us  now  examine  the  terms  of  the  second  order. 
We  have 


Now  when  x  decreases  algebraically — that  is,  when  the  projec- 
tile is  ascending — we  must  regard  the  velocity  as  negative. 
But  then  ds  is  also  negative,  so  that,  both  radicals  in  (8)  be- 
coming negative,  SU  wiW  be  positive.  When  x  increases — that 
is,  when  the  projectile  is  descending — both  radicals  become 
positive,  so  that  SU'is  positive. 

If,  then,  the  arc  of  the  parabola  with  which  we  are  con- 
cerned does  not  include  the  vertex,  we  undoubtedly  have  a 
minimum ;  but  if  we  are  required  to  reach  or  pass  beyond  the 
vertex,  then,  since  y  at  that  point  becomes  infinite,  our  conclu- 
sion that  we  shall  have  still  a  minimum  cannot  be  regarded  as 
altogether  trustworthy,  and  we  shall  be  obliged  to  resort  to 
another  method  of  investigation. 


278  CALCULUS  OF  VARLATIONS. 

214.  Let  us  now  assume  the  horizontal  as  the  axis  of  x, 
estimating  y  vertically  downward,  and  taking  the  origin  at  the 
vertical  distance  a  above  the  first  fixed  point.  Then  we  shall 
have 

,  u=iywTr)dx=fydx.        (9) 

Whence,  by  formula  (C),  Art.  56,  we  have 

Hence 

so  that 
Whence 


x=  ±2C,Vy-C,  +  C, 


y=C^  +  ^—-^.  (10) 

Differentiating  (10),  we  have 

/  =  ^'.  (n) 

Now  (10)  is  the  equation  of  a  parabola  when  the  directrix 
is  taken  as  the  axis  of  x,  the  origin  being  assumed  at  pleasure, 
and  C^  is  the  abscissa,  while  C^  is  the  ordinate  of  the  vertex, 
46^1  being  twice  the  parameter,  or  2p.  For  making  y'  zero  in 
(i  i),  we  have  x  =  Q,  and  then  (10)  gives,  for  the  same  point  of 
the  curve,  y  =  C,.  Now,  changing  the  origin  to  the  point 
Q,  C^,  we  shall  obtain,  after  interchanging  the  variables  x 
and  y,  y"  =  a^C^x  =  2px.  Hence  we  see  that  the  distance  of 
the  directrix  above  the  starting-point  is  always  numerically 
equal  to  a,  or  the  height  due  to  the  initial  velocity. 


DISCONTINUOUS  SOLUTIONS.  279 

Now  we  know  that  the  focus  of  any  parabolic  path  de- 
scribed by  a  projectile  moving  from  ^  to  ^  must  be  at  the 
intersection  of  two  circular  arcs,  described  with  the  same 
radius  a  from  the  two  points  respectively  as  centres.  But  if 
a  be  so  assumed  that  these  circles  cannot  touch,  there  can  be 
no  continuous  solution,  and  the  point  B  will  be  without  the 
limiting  parabola.  If  the  circles  touch,  one  parabola  can  be 
drawn,  having  its  focus  upon  the  line  AB,  the  point  B  being 
then  upon  the  limiting  parabola ;  while  if  the  circles  intersect, 
there  will  be  two  parabolic  paths  along  which  the  particle 
may  move,  the  first  having  its  focus  below,  and  the  second 
above  the  line  AB^  the  point  B  being  in  this  case  within  the 
limiting  parabola. 

215.  It  will  be  seen  that  by  changing  the  independent 
variable  we  avoid  any  infinite  value  of  y,  and  we  will  now  pro- 
ceed to  show  that  when  the  parabolic  arc  has  its  focus  below 
the  line  AB,  the  action  becomes  a  minimum,  but  that  when 
the  focus  is  upon  or  above  AB,  the  action  is  not  a  minimum. 

Employing  Jacobi's  method,  we  have,  from  (9), 


or     -^       — 


dr       ""    virryy 


(13) 


which  is  always  positive,  and  remains  finite  throughout  the 
range  of  integration,  so  that  we  shall  have  a  minimum  if  we 
can  take  u  so  that  it  shall  not  vanish  within  the  same  range, 
and  that  u^  may  remain  finite.     From  (10)  we  have 

the  value  of  y'  being  taken  from  (11).  Therefore  the  most 
general  value  of  u  is 

u=i- /'  -  Ly\         and     u'  =  -  2//'  -  Ly\        (15) 


28o  CALCULUS   OF  VARIATIONS. 

Now  because  y'  and  y"  remain  finite,  u'  will  not  become  infi- 
nite ;  and  to  make  u  vanish,  we  must  have 

L=^,-y=-H,  (i6) 

and  we  shall  have  a  minimum  if  the  range  of  H  over  all  real 
values  be  only  partial,  but  none  if  it  be  complete. 

216.  In  order  that  H  may  range  over  all  real  values,  it 
must  certainly  touch  zero  and  infinity.  The  first  condition 
requires  7'  to  become  ±1,  and  is  fulfilled  at  both  extremities 
of  the  latus  rectum,  and  there  only.  The  second  requires  y' 
to  become  either  zero  or  infinity,  the  latter  condition  being 
never  fulfilled,  and  the  former  at  the  vertex  only.  Now  let 
y^'  and  y^  be  the  values  of  y'  at  the  extremities  of  any  focal 
chord.  Then,  because  the  tangents  to  the  parabola  at  these 
extremities  meet  at  right  angles  upon  the  directrix,  we  must 

have  y'  = „  and  hence  we  shall  find  that 

y:---r     or     H,^y:-~     or     H,. 

)\  y^ 

Therefore  as  H  in  this  case  starts  with  a  certain  value,  changes 
sign  by  passing  through  infinity  at  the  vertex,  and  returns  to 
its  initial  value,  its  range  must  be  at  least  complete,  and  we 
have  not  a  minimum.  If  the  arc  were  still  greater,  the  range 
of  //"would  be  more  than  complete. 

Now  H  can  return  to  its  initial  value  but  once,  although 
it  may  pass  that  value.  When  the  initial  value  is  zero,  this  is 
evident,  since,  as  we  have  seen,  there  is  but  one  other  point  at 
which  H  can  be  zero.  When  the  initial  value  is  not  zero,  // 
must  change  sign  twice  before  returning  to  its  initial  value, 
and  four  times  before  returning  to  it  a  second  time,  and  this 
latter  is  impossible,  since  there  are  but  three  points  at  which 
H  can  change  sign  at  all. 


DISCONTINUOUS  SOLUTIONS.  28 1 

Since,  then,  the  values  of  H  at  the  extremities  of  any  focal 
chord  are  equal,  they  will  be  equal  nowhere  else,  and  the 
range  of  H  is  then  just  complete. 

If,  therefore,  the  arc  in  question  be  less  than  that  subtended 
by  a  focal  chord,  the  range  of  H  is  not  complete,  and  the  action 
becomes  a  minimum.  In  other  words,  we  see  that  the  action 
will  not  be  a  minimum  unless  the  second  fixed  point  be  so  situ- 
ated that  tw^o  parabolic  arcs  are  admissible,  and  then  for  that 
path  only  which  has  its  focus  below  the  line  AB. 

217-  Since  there  can  be  no  continuous  solution  when  the 
second  fixed  point  lies  on  or  without  the  limiting  parabola,  we 
next  inquire  whether  there  ma}^  not  be  some  discontinuous 
solution  or  solutions  in  these  cases. 

We  first  ask,  then,  whether  we  have  unconsciously  imposed 
any  boundary  along  which  the  sign  of  Sy  is  fettered ;  because,  if 
not,  the  solution  can,  at  least  so  far  as  discoverable  by  the  cal- 
culus of  variations,  consist  only  of  some  combination  of  lines 
satisfying  the  equation  M  —  o.  But  we  see  from  (lo)  that  y  =  o 
is  such  a  boundary,  since  to  make  y  negative  would  render  the 
velocity  imaginary,  and  with  the  notation  of  (i)  this  boundary 
is  given  by  the  equation  x-\-  a=^o. 

218.  Let  us  next  see  what  can  be  obtained  from  the  funda- 
mental equations  given  by  the  two  methods  previously  em- 
ployed.    These  are 

because  there  is  no  escape  from  these,  if  we  make  M  vanish  in 
each  case.  The  first  of  these  equations  is  satisfied  by  y  ==  o 
and  ^  =  o,  and  alsoy  —  oo ,  because  in  the  latter  case  we  ob- 
tain X  -\-  a^zc. 

Passing  for  the  present  the  question  of  combining  y  =  o 
and  y  ==  00 ,  it  is  suggested  that  our  solution  may  consist,  m 


282 


CALCULUS  OF  VARIATLONS. 


part,  of  some  line  parallel  to  the  axis  of  y.  But  because  y' 
would  here  become  infinite,  we  cannot,  while  keeping  the  ver- 
tical as  the  axis  of  x,  investigate  the  variation  of  C/ along  this 
line.  But  the  second  of  equations  (17),  in  which  the  axis  of  x 
is  horizontal,  offers  the  same  solutions  as  the  first,  since  it  is 
satisfied  by  j'  =  00  ,  6^1  =  o,  or  by  y'  =  o,  which  gives  y  =  C^j 
which  is  the  same  condition  as  was  before  expressed  by 
X  -\-  a  =  o. 

With  this  change  of  the  independent  variable,  we  can  ex- 
amine the  condition  which  we  were  before  unable  to  investi- 
gate ;  namely,  whether  the  solution  may  be  composed,  in  part, 
of  some  horizontal  line. 

Now  if  this  horizontal  be  any  other  than  the  boundary 
/  =  o,  it  must,  since  along  it  ^y  is  of  unrestricted  sign,  satisfy 
the  equation  M  =  o.  But  this  equation,  when  C/"has  the  form 
given  in  (10),  becomes 


M  = 


2  Vj  dx  ^\^  y' 


(18) 


But  when  we  puty  =  o  and  y  =  C^,  we  have  M=^  - — _:  and 

2  Vl, 

as  this  does  not  vanish,  this  solution  must  be  abandoned. 


2(9.  As  the  horizontal  line  j/  =  o  is  not  yet  known  to  be 
excluded,  since  it  need  not  satisfy  the  equation  M=^o,  and  as 


y'  z=i  00  was  also  suggested  as  a  solution,  it  remains  to  consider 
whether  the  solution  may  not  be  found  by  combining  this  hori- 


DISCONTINUOUS  SOLUTIONS.  283 

zontal  with  the  verticals  through  the  two  fixed  points,  as  in 
the  figure,  where  the  path  of  the  projectile  is  supposed  to  be 
ACDB. 

Of  course  a  particle  could  not  move  from  A  to  B  along  this 
broken  line,  because  its  velocity  along  CD  would  become  zero. 
But  we  can  draw  a  curve  indefinitely  near  to  A  CDS  along 
which  the  velocity  will  not  become  exactly  zero,  and  then  we 
shall  find  that  the  action  along  this  curve  will  be  greater  than 
that  along  the  discontinuous  path. 

To  determine  whether  the  line  ACDB  is  the  path  of  mini- 
mum action,  we  shall,  on  account  of  the  infinite  value  of  y' , 
need  some  other  method  of  investigation,  and  we  might  try 
transforming  to  polar  co-ordinates.  Still  an  analytical  demon- 
stration will  not  here  be  necessary.  For  let  AC  and  At  be 
equal  in  length,  and  let  them  be  divided  into  the  same  number 
of  equal  and  infinitesimal  parts ;  and  let  PQ  and  pq  be  a  cor- 
responding pair,  so  that  AP  will  equal  Ap.  Then  because  P 
is  vertically  higher  than  /,  the  velocity  at  P  will  be  less  than 
that  at/,  and  the  action  through  PQ  less  than  that  through/^. 
Hence  it  appears  that  the  entire  action  through  ^(T  is  less  than 
that  through  Ac,  In  like  manner  we  show  that  the  action 
through  BD  is  less  than  that  through  Bd.  Now  the  action 
along  CD  is  zero,  while  that  along  cd  is  not ;  so  that  it  is  cer- 
tain that  the  action  along  the  primitive  A  CDB  is  less  than  that 
along  the  derivative  AcdB^  even  if  we  do  not  vary  the  bound- 
ary CD,  and  much  more  so  if  we  vary  that  line. 

220.  It  is  easy  to  see  that  the  discontinuous  solution  which 
we  have  obtained  is  admissible  even  when  the  parabolic  path 
also  renders  the  action  a  minimum.  When  the  second  fixed 
point  lies  on  or  without  the  limiting  parabola,  the  discontinu- 
ous solution,  being  the  only  one  which  presents  itself,  undoubt- 
edly renders  the  action  the  least  possible,  as  well  as  a  minimum. 
When  both  minima  are  admissible,  we  shall  find  that  some- 
times the  one  and  sometimes  the  other  will  give  the  smaller 


284  CALCULUS  OF  VARIATIONS. 

minimum ;  and  there  can  be  little  doubt  this  smaller  minimum 
is  in  every  case  the  least  possible  value  also  of  the  action. 

The  comparison  of  the  two  minima,  when  they  exist,  must 
be  effected  by  the  ordinary  calculus,  but  we  subjoin,  without 
proof,  the  necessary  formulae.  (See  Todhunter's  Researches, 
Art.  173.) 

Let  g  be  the  force  of  gravity,  r^  and  r,  the  radii  vectores  of 
the  two  fixed  points,  C  the  length  of  the  chord  joining  these 
points,  and  w  the  action.  Then  for  the  parabolic  path,  accord- 
ing as  it  subtends  less  or  more  than  two  right  angles  at  the 
focus,  we  shall  have 


or 


w  =  :^i(^  +  ^  +  0«+  ('-0+'-.  -  Cf\- 


(19) 


For  the  discontinuous  solution  the  action  is  that  due  to  pass- 
ing along  the  verticals  only,  and  is 

^=^(..+..).  (20) 

221.  The  principles  which  have  been  previously  explained 
regarding  the  origin  and  nature  of  discontinuous  solutions  are 
equally  applicable  when  polar  co-ordinates  are  employed,  and 
we  shall  find  in  thi-s  case  also  that  they  are  generally  in  some 
manner  presented  as  a  solution  of  the  equation  J/  =  o,  although 
they  may  not,  and  need  not  always,  really  satisfy  that  equation 
at  all.     Let  us  now  briefly  consider  a  problem  of  this  kind. 


DISCONTINUOUS  SOLUTIONS.  285 


Problem   XXXVI. 

It  is  required  to  determine  whether  there  be  any  disco7itinnoiis 
solution  involved  in  Prob.  XXII. 

We  have  seen,  Art.  123,  that  when  the  second  fixed  point 
lies  without  a  certain  limiting  ellipse,  no  elliptic  arc,  satisfy- 
ing all  the  conditions  of  the  question,  can  be  drawn  between  it 
and  the  first  fixed  point ;  and  that  even  when  it  is  situated  on 
the  limiting  ellipse,  and  there  can  be  one  eUipse  drawn,  it  does 
not  render  the  action  a  minimum.  It  appears,  then,  that  if 
there  be  any  solution  in  these  cases,  it  must  be  discontinuous ; 
and  the  analogy  of  the  last  problem  would  lead  us  to  expect, 
what  is  indeed  the  fact,  that  even  when  a  continuous  solution 
exists,  a  still  smaller  value  of  the  action  is  in  some  cases  given 
by  a  certain  discontinuous  solution. 

222.  Now  the  fundamental  equation  of  this  problem  is 
equation  (8), 

Wr"" 

where 


W^  i/--i,  (2) 

r        a  ^  ' 

and  we  cannot  help  arriving  at  this  equation  if  we  make  M 
vanish.  But  if  in  (i)  we  make  c  zero,  that  equation  will  be 
satisfied  by  /  =  ^  or  W—  o.  The  first  would  indicate  that 
we  might  employ  some  portion  of  the  radius  vector  drawn  to 
one  or  both  the  fixed  points,  or  of  these  radii  produced. 
To  interpret  the  second  we  have 

W'  =  ---  =  o, 
r       a 

so  that 

I        I 

-  =  — ,         r  =  2a. 
r        2a 


286  CALCULUS  OF    VARIATLONS. 

That  is,  it  is  suggested  that  a  portion  of  the  solution  might 
consist  of  a  circular  arc  described  from  the  centre  of  force 
with  a  radius  2a. 

Let  O  be  the  centre  of  force,  A  and  ^the  two  fixed  points. 
Then  the  discontinuous  solution  which  is  proposed  is  the  path 
A  CDB,  where  CD  is  the  portion  of  the  above-named  circular 
arc  intercepted  between  OA  and  OB  produced. 

But  before  considering  whether  the  proposed  solution  does 
render  the  action  a  minimum,  we  inquire  whether  any  bound- 
ary exists  along  which  the  sign  of  Sr  is  fettered,  and  which 
need  not  therefore  satisfy  the  equation  J/ =  o.  Now  the  value 
of  the  velocity  v' ,  equation  (4),  Art.  121,  is 


\/^l--l=WVf,  (3) 


where  /  is  the  intensity  of  the  attracting  force  at  a  unit's  dis- 
tance. When,  therefore,  VV  vanishes  and  r  becomes  2^,  v' 
becomes  zero ;  and  when  we  make  r  greater  than  2a,  v'  be- 
comes imaginary.  Hence  the  arc  CD  is  itself  such  a  boundary, 
unconsciously  imposed,  and  along  it  ^r  must  be  negative  and 
the  action  zero. 

223.  Owing  to  the  infinite  value  of  /,  we  cannot  deter- 
mine, by  adhering  to  polar  co-ordinates,  whether  the  pro- 
posed solution  Avill  render  the  action  a  minimum  or  not,  and 
the  natural  mode  of  procedure  would  be  to  express  the  value 
of  dU\n  rectangular  co-ordinates,  by  which  we  could  escape 
infinite  values.  But  this  will  not  be  necessary,  because,  by 
reasoning  precisely  similar  to  that  employed  in  Art.  219,  it 
will  appear  that  the  action  through  A  C  and  DB  must  be  less 
than  that  through  any  curve  of  the  same  length  which  can  be 
derived  by  the  method  of  variations,  and  the  arc  CD  cannot 
be  reached  by  curves  which  do  not  exceed  these  lines.  Then 
as  the  action  is  zero  along  CD^  it  is  evident  that  the  discon- 


DISCONTINUOUS   SOLUTIONS.  28/ 

tinuous  path  A  CDB  will  render  the  action  less  than  would  any 
other  path  which  could  be  derived  from-  it  by  the  calculus  of 
variations. 

Problem  XXXVII. 

224-.  A  steamer  is  to  pass  from  one  port  to  another  on  a 
stream  whose  current  flows  always  in  the  same  direction,  Jier  speed 
beiitg  dependent  solely  upon  the  angle  which  Jier  course  makes  with 
the  directio7i  of  the  current,  together  with  certain  constant  quanti- 
ties. It  is  required  to  determine  the  fo^m  of  her  path,  so  that  the 
passage  may  be  made  in  the  shortest  time  possible. 

Assume  the  course  of  the  current  as  the  axis  of  x,  and  esti- 
mate X  in  the  direction  of  its  flow.  Also  let  v  be  the  velocity, 
t  the  time,  and  ds  an  element  of  the  required  path.  Then 
since  v  depends,  in  some  fixed  manner,  upon  constants  and 
the  angle  between  the  path  and  the  axis  of  x,  we  may  write 
V  =  F{y'),  and 


ds        VI  -\-y'\lx  ,    ,     y  J 

dt  —  -   —  ^-^- —  /( J  )  dx  =  fix, 

V  r 

Hence  the  expression  to  be  minimized  is  ^=  /     fdx,  where 

t/  Xq 

it  is  evident  that /can  become  any  function  whatever  oi  y'. 

Now  we  have  already  seen.  Art.  56,  that  the  solution  of 
this  problem  is  given  always  by  a  straight  line,  and  there  is 
no  escape  from  this  conclusion  so  long  as  we  make  M  vanish. 
For 

dx  dy  dx 

so  that  if  M be  zero,  we  cannot  help  obtaining  f  —  c\  and  to 
satisfy  this  equation,  y'  must  certainly  be  a  constant,  which 
w^ill  lead  to  a  right  line  as  the  only  possible  solution.  But 
since  the  required  line  is  in  this  case  to  pass  through  two  fixed 


288  CALCULUS  OF    VARIATLONS. 

points,  we  seem  at  first  to  be  restricted  to  a  single  course  for 
all  possible  conditions,  whereas  a  little  reflection  will  serve  to 
show  us  that  we  could  easily  impose  such  conditions  as  would 
enable  us  to  shorten  the  time  of  passage  by  pursuing  a  path 
not  always  coinciding  with  the  straight  line  joining  the  two 
points. 

It  appears,  however,  upon  examination,  that  the  equation 
M—o  must  hold  throughout  the  entire  course,  as  we  cannot 
find  that  any  boundary  has  been  in  any  way  imposed  along 
which  Sy  or  dy'  will  be  of  restricted  sign.  We  feel  certain, 
therefore,  that  no  solution  can  be  obtained,  at  least  by  the  cal- 
culus of  variations,  except  a  right  line,  or  one  composed  of 
right  lines.  But  since  f  is  a  constant,  suppose  that  constant 
to  become  zero.  Then  if  the  equation  f'-=o  furnish  more 
than  one  real  value  of  y\  we  may  have  two  or  more  lines 
meeting  at  finite  angles.  For  the  terms  free  from  the  sign  of 
integration,  which  are  f^'^y^  —  fj^y^  +  etc.,  would  vanish, 
because  f  would  vanish  for  either  of  the  meeting  lines, 
although  the  values  oi  y'  for  the  two  lines  might  differ.  We 
shall,  however,  illustrate  this  problem  by  considering  some 
particular  cases. 

225.  1st.  Let  a  be  the  angle  between  the  path  and  the  axis 
of  X,  which  is  not  to  exceed  — ,  and  suppose  the  velocity  v  to 

vary  as  cos  a  = =  .     Then  in  this  case   U  be- 

sec  a       4/i._Ly2 

comes  U=£\i  ^y')dx  =fj^'fdx,   giving  f  ^  J-  =  2/. 

Now  y'  must  have  the  same  value  throughout  the  integral, 
because  if  it  change  value*  at  any  point  x^,  j/^,  we  shall  have, 
as  already  explained,  without  the  integral  sign,  after  trans- 
forming the  term  of  the  first  order  in  the  usual  way, 

f:^y.-f:^y..     or     2{^y:dy,-y:dy^,     or     2(j// -X^Jo, 


DISCONTINUOUS  SOLUTIONS.  289 

which  must  vanish,  since  Sy^  may  have  either  sign.  Hence,  in 
this  case,  the  minimum  time  will  be  gained  by  following  the 
right  line  joining  the  two  points ;  and  because  only  one  value 
of  y  is  admissible,  we  infer  that  this  path  gives  also  the  least 
value  of  /,  t  being  certainly  a  minimum,  since  the  term  of  the 

second  order  is  /  Sy'^dx.  In  this  case,  then,  there  is  no  dis- 
continuity, but  we  now  pass  to  an  example  in  which  it  occurs. 

226.  2nd.     Let 


2^4 


so  that 


/=-?+f         <.) 


where  b  is  some  constant.   Then  proceeding  as  usual  with  the 
integral  U=  J     fdx,  we  obtain 

/'=/(/'- 1) =^-  (2) 

Whence  we  also  find 

P-/"-3/'-i.  (3) 

Now  if  we  solve  (2)  without  restriction,  we  shall  obtain 
a  straight  line,  which  must  of  course  pass  through  the  two 
fixed  points,  and  we  will  first  examine  whether  this  continuous 
solution  will  always  render  the  time  of  passage  a  minimum. 
Now  since  the  term  of  the  second  order  in  6U  is 


-jy'^y"'^-^ 


2^^o 


290  CALCULUS   OF  VARLATIONS. 

we  shall  have  a  maximum  or  a  minimum  according  as  f"  is 
negative  or  positive.      Hence,  from  (3),  observing  that  tan^ 

—  =  —  ,  we  see  that  when  the  angle  is  less  than  — ,  the  time  is 
63  6 

a  maximum ;  but  that  if  the  ports  were  so  situated  that  the 

line  joining  them   must  make   with  the  axis  of  x  an  angle 

greater  than  30°,  the  time  will  become  a  minimum. 

227.  Now  when  we  have  shown  t  to  be  in  any  particular 
case  a  maximum  or  a  minimum,  it  does  not  follow  that  we 
have  obtained  its  greatest  or  least  value,  since  some  discon- 
tinuous solution  may  give  a  greater  maximum  or  a  smaller 
minimum.  Now  if  there  be  any  discontinuous  solution,  it 
must  cause  f  or  c  in  (2)  to  retain  the  same  value  throughout 
U,  otherwise  there  would  arise  terms  of  the  form  {f^' —fj)^y^, 
which  would  not  vanish.  Any  values,  then,  of  y'  which  will 
satisfy  the  equation  /'  =  c,  in  which  we  may  give  to  c  any 
value  we  please,  only  retaining  the  same  throughout  U,  may 
be  combined  into  one  solution,  provided  this  combination  will 
enable  us  to  pass  from  one  fixed  point  to  the  other,  and  pro- 
vided also  that  the  various  parts  of  the  combination  do  not 
render  the  terms  of  the  second  order  of  variable  or  conflict- 
ing sign. 

Suppose,  in  the  present  case,  we  make  c  zero.  Then 
we  obtain,  as  the  roots  of  (2),  /  =  o,  /  —  i,  /  =  —  i.  But 
the  last  two  values  of  /  render  f  in  (3)  positive,  while  the 
first  renders  it  negative,  and  cannot,  therefore,  enter  any  solu- 
tion with  the  other  two,  as  the  sign  of  the  terms  of  the  second 
order  would  then  be  in  our  power.  It  is  evident  that  a  vessel 
could  pass  from  one  point  to  any  other  by  a  suitable  combina- 
tion of  tacks,  making  with  the  axis  of  x  angles  whose  tangents 
are  either  -|-  i  or  —  i  ;  and  as  the  integral  has  the  same  value, 
whatever  be  the  number  of  these  tacks,  because /is  the  same 
whether  y  be  +  i  or  —  i,  we  obtain  in  all  cases  one  path 
along  which  the  time  of  passage  will  be  a  minimum. 


DISCONTINUOUS  SOLUTIONS  29 1 

To  determine,  when  two  minima  exist,  Avhether  the  quicker 
passage  can  be  made  by  following  the  path  composed  of  tacks 
or  a  continuous  line,  is  not  a  problem  of  variations,  but  of 
algebra  only.     For,  resuming  the  value  of/,  we  may  write 

24       4 '    4 

Also,  when  y'  is  +   i  o^*  —  i>  /=  <^' •      Therefore,  since 

4 
(y^  —  i)'  cannot  become  negative,  wx  see  that  the  solution 
composed  of  tacks  will  give  the  least  possible  value  of  t.     We 
have,  of  course,  assumed  that  it  is  not  necessary  to  tack  back- 
ward ;  that  is,  that  x  may  always  increase  algebraically. 

228.  We  naturally  inquire  whence  arises  the  discontinuity 
in  this  class  of  problems,  and  why  it  presents  itself  in  certain 
forms  of  /,  and  not  in  others.  Now  the  only  condition  im- 
posed besides  the  fundamental  one,  that  the  given  line  shall 
possess  a  certain  maximum  or  minimum  property,  is  that  it 
shall  also  join  two  fixed  points,  and  if  the  required  maximum 
or  minimum  property  be  not  altogether  impossible,  the  dis- 
continuity must  result  from  imposing  the  second  condition. 
That  it  does  in  general  thus  arise  will  appear  from  the  fol- 
lowing example,  in  which  this  condition  is  removed. 


Problem  XXXVIII. 

229.  A  vessel  starting  from  a  fixed  point  is  required  to  sail  a 
certain  number  of  miles,  her  speed  being  ahvays  dependent  solely 
upon  the  direction  of  her  course  and  certain  constant  quantities. 
It  is  required  to  determine  along  what  path  the  given  distance 
may  be  accomplished  in  a  minimum  time. 

Regarding  the  ocean  as  a  plane,  assume  the  meridian 
through    the    starting-point   as   the   axis   of  -3{;i,.  and    employ 

y^''      ^  — *  J-.    >.->■ 


292  CALCULUS  OF  VARIATIONS. 

t  and  V  as  before.     Then  it  is  plain  that  we  shall  have,  as  for- 

4/1  _u/-^ 

merly,  v  —  F{y')  =  F,  and  dt  = ^^^— -  dx  =  /{/)  dx  =  fdx. 

r 

Hence    we   are    to    minimize   the    expression   /    fdx,    while 

nx-i,     

/      \/i-\-y'''dx  is  to  remain  constant.      Therefore  the  prob- 

fJ  Xq 

lem  is  now  one  of  relative  maxima  and  minima,  and  we  have 

u = fC{f+ ^  ^^+yi  ^--  =£?  vdx,      (I) 

where  it  must  be  observed  that  V  is  also  a  function  of  y'  and 
constants  only.  In  the-  present  case,  moreover,  we  do  not 
suppose  the  second  extremity  of  the  required  curve  to  be  in 
any  manner  restricted,  so  that  x^  and  y^  are  both  variable. 
Therefore,  to  the  first  order,  we  have 

SU^  (/+  a  ^7  +  /')  dx,  +  j/'  +  -^7=^  I  Sy, 


Whence 


where 


rij/'+vf$7-i^-""-      « 


-^     df 


Now,  for  the  same  reason  as  given  in  the  preceding  prob- 
lem, c  cannot,  even  should  discontinuity  occur,  and  the  inte- 
gral be  separated  into  parts,  have  two  values,  c^  and  ^3,  within 
the  range  of  integration ;  and  since  we  know  from  (3)  that  c,  or 
the  coefficient  of  Sy^,  must  vanish,  (3)  becomes 


DISCONTINUOUS   SOLUTIONS.  293 

Moreover,  in  this  case,  no  relation  exists  between  dx^  and 
6>^,  because  the  extremity  of  the  required  curve  is  not  con- 
fined to  any  other  curve,  but  is  wholly  unrestricted.  There- 
fore (i)  must  also  give 

(/+«vr+7^X  =  o.  (5) 

From  (4)  and  (5)  we  have 


/     \  and    ,^-r^;+y'.         (6) 


vi+yM,  y 

From  (4)  and  (6)  we  obtain 

Now  since  F  is  a  function  of  y  only,  we  know  that  the 
required  path  must  be  some  right  line,  or  combination  of 
right  Imes,  so  that  y  is  the  tangent  of  the  inclination  of  this 
line,  or  else  of  the  last  tack,  to  the  axis  of  x.  But  it  is  evi- 
dent that  if  the  solution  can  consist  of  tacks,  involving  two  or 
more  values  of  y ,  the  arrangement  of  these  tacks  will  be  arbi- 
trary, since  the  integral  taken  through  an}'  given  portion  of 
X  will  be  the  same  for  any  one  of  the  tacks — that  is,  for  any 
one  of  the  admissible  values  of  y — and  therefore  //  can  have 
any  one  of  these  values,  but  no  others.  Hence,  as  the  pos- 
sible values  of  y  and  7/  are  the  same,  we  may  remove  the 
suffix  from  (7)  and  write,  as  the  general  equation  of  condition, 

/'-T^  =  o.  (8) 

From  (8)  we  can  obtain  y  in  terms  of  constants  only,  and 
it  may  have  one  or  more  real  values,  the  imaginary  roots  being 
of  course  rejected.  In  the  former  case  there  can  be  but  one 
solution  ;  but  when  y  has  more  than  one  real  value,  a  discon- 


294 


CALCULUS  OF   VARLATIONS. 


tinuous  solution  by  a  combination  of  these  values  would  seem 
possible.  It  must,  however,  be  observed  that,  whether  the 
solution  be  continuous  or  not,  a  must  retain  the  same  value 
throughout  U. 

Now  take  any  two  real  values  of  y'  found  from  (8),  and 
make  //  equal  to  the  first,  and  7/,  which  may  be  regarded  as 
measuring  the  slope  of  some  other  tack,  equal  to  the  second. 
Then  from  (6),  and  also  observing  that  we  may  interchange 
the  slopes  of  the  tacks  at  pleasure,  we  have 


f 


Vi+y 


(9) 


and  as  every  member  in  (9)  equals  —  a,  we  may  write 


/ 


Vi  +/■' 


A-r\ 


_^      / 


^i  +/" 


y' 


(10) 


But  it  will  be  in  general  impossible  to  satisfy  (10)  by  employ- 
ing any  two  values  of  y'  found  from  (8),  so  that  a  discontinu- 
ous solution  will  not  frequently  occur.  Still  such  solutions  are 
possible,  as  we  shall  prese^ntly  show ;  and  even  when  no  dis- 
continuity is  admissible,  it  is  conceivable  that  we  may  have  a 
choice  of  two  continuous  solutions,  provided  j//  and  //  can 
severally  satisfy  the  equations 


/ 


/ 


DISCONTINUOUS  SOLUTIONS.  2g$ 

because  a  in  the  two  solutions  need  not  be  identical,  but  must 
not  change  value  in  the  same  solution. 

230.  As  a  particular  example  of  the  preceding  problem, 
let  us  assume  the  velocity  to  be  that  employed  in  case  2nd, 
Prob.  XXXVII.,  so  that  /and  /'  will  have  the  values  there 
given.  Then  by  equations  (i)  and  (2),  Art.  226,  equation  (8) 
of  the  preceding  article  will  become 


O.  (I) 


/(/'- 

--"'+ 

± 

i)  —  f  — 

.+/■ 

Therefore, 

if  y  be  not 

zero,  we 

have 

/'-I 

-1- 

2     4 

/i 

—  o 

I  2      •     4    ) 

or 


/'  +  ^'=  4(^1+11.  (2) 

3  3  ' 


Whence 

^             3       '1^^121 

3 

(3) 


Now  if  VB  be  less  than  -,  /  will  always  be  imaginary ;  if  it 

equal  -,  /  will  be  zero ;  while  if  it  exceed  -,  one  of  the  values 

3^  3 

of  y  will  be  negative,  and  all  the  real  values  of  y'  will  be 
given  by  the  equation 


y^±V---i.VB.  (4) 

3 


Now  we  have 


and 


CALCULUS  OF    VARIATIONS. 


2    "^   4 


vi  +y'       Vi  +y' 


fWi+y 


=  {y'-i)Vi+y^ 


is) 


But  it  at  once  appears  that  none  of  the  members  of  (9)  will  be 
in  any  way  affected  by  the  successive  substitution  of  two 
values  of  y  numerically  equal  but  with  contrary  sign.  More- 
over, in  this  case,  the  equation 


/     _rvi+y 


Vi  +y'        y 

reduces  to  equation  (2) ;  so  that  it  must  be  satisfied  by  either 
value  of  y  just  found,  and  will  also,  from  what  has  been  shown, 
be  satisfied  by  substituting  the  positive  value  in  one  member 
and  the  equal  negative  value  in  the  other. 

Hence  it  appears  that  equation  (10)  of  the  preceding  article 
will  be  satisfied  by  putting  for  j//  and  j/  the  two  values  of  y 
given  in  (4),  and  by  no  others.  Therefore  the  solution  y  =  o 
can  only  hold  when  zero  is  a  root  of  (2),  which  can  only  be 
made  true  by  making  d^  =  —  i,  and  in  this  case  there  will  be 
no  other  root,  and  so  no  discontinuity.  But  if  d^  become 
greater  than  —  i,  we  shall  have  an  equal  positive  and  nega- 
tive value  of  y,  which  may  be  combined  in  the  same  solution, 
thus  giving  discontinuity. 

231.  Let  us  now  consider  the  terms  of  the  second  order. 
These  are 


where 


'''-i{T)f^'+'''''^'+ir'£^'y''^'    (6) 


2      4 


DISCONTINUOUS  SOLUTIONS.  297 

Now,  because  y"  is  zero,  all  the  successive  differential  co- 
efficients of  V^  must  vanish,  and  also  we  have 


=W-'^^\''- 


and,  as  will  be  seen  from  equation  (4),  Art.  229,  the  coeffi- 
cient of  (^j//  likewise  vanishes,  so  that  we  have  left  in  SU  only 
the  terms  under  the  integral  sign,  and  have  merely  to  deter- 
mine the  sign  of  -—^.     Now  we  have 

^=  /"  +       ^  /"  =  3/'  -  I, 


/Vi+y 


a  ^  y  =  -  (/'  -  I)  Vi  +/'. 


Whence 


^^^_3y._i_y^-i_3/*+y 


dy"       ^-^  I  +/'         I  +/'  * 


Therefore  it  appears  that  we  have  a  minimum  whether  y'  be 

positive  or  negative. 

When,  however,  y'  is  zero,  we  see  from  the  last  equation 

d'^V 
that  -r-TT  is  also  zero,  so  that  we  might  infer  that  this  value  of 
dy 

y  gives  neither  a  maximum  nor  a  minimum.  But  this  infer- 
ence would  not  in  the  present  case  be  correct,  because  we 
shall  find  that  the  terms  of  the  third  order  reduce  also  to  zero, 
while  those  of  the  fourth  order  will  become  positive.  It  may 
be  also  observed,  although  not  affecting  the  problem,  that 


298  CALCULUS  OF    VARIATIONS. 

when  y  is  zero,  dx^  can  have  but  one  sign,  the  negative,  if  x^ 
be  positive. 

232.  It  will  be  seen  that  while  the  removal  of  all  condi- 
tions regarding  the  upper  limit  does  not  here  destroy  the  ad- 
missibility of  a  discontinuous  solution,  it  nevertheless  abol- 
ishes its  necessity.  For  as  the  value  of  /,  and  also  that  of  x^, 
will  be  the  same  whether  we  employ  the  positive  or  the  nega- 
tive value  of  /',  or  some  combination  of  the  two,  the  time 


fdx  will  be  also  unaltered  ;  and  as  we  are  not  now  obliged 

to  tack  in  order  to  go  from  one  fixed  point  to  another,  and  no 
time  is  gained  by  tacking,  the  discontinuity  is  merely  admis- 
sible. The  discontinuity  in  this  case  appears  to  arise  from  the 
fact  that  the  problem  is  so  constructed  that  the  fundamental 

equation  f  -\ „  -^ =  c  may  have  two  roots,  both  of  which 

1/1  +y" 

give  the  same  value  of  /,  and  satisfy  all  the  conditions  of  the 
question. 

233.  Suppose  we  modify  the  preceding  example  byre- 
quiring  that,  instead  of  sailing  a  certain  number  of  miles,  the 
vessel  shall  be  required  to  reach  a  certain  degree  of  latitude 
in  a  minimum  time.     Then  we  are  to  minimize  absolutely  the 

expression  U  —  J  fdx,  where  /  has  the  same  value  as  be- 
fore, the  limit  x^  now  being  fixed,  but  jj  being  subject  to  varia^ 
tion.  Then  we  havej  as  before,  f  =  c,  and  c  cannot  have  two 
values.  But  because  ^y^  is  not  zero,//  ot  f  must  vanish,  so 
that  we  have/^  =  y'  (y^  —  i)  =  o ;  the  roots  of  which  arey  =  o, 
y'—  i,y'—  —  I.  Now  asji'  is  not  fixed,  we  can  employ  any 
one  of  the  values  of  y'  alone  throughout  U.  The  first  will 
render  U  a  maximum,  as  we  have  already  seen,  while  the 
other  two  will  give  ^the  same  value  whether  employed  sep- 
arately or  in  combination,  which  value  is  a  minimum,  as  has 
been  shown,  and  is  also  the  least  value  of  U, 


DISCONTINUOUS  SOLUTIONS.  299 

234.  We  may  now  consider  briefly  the  inquiry  with 
which  we  opened  Art.  228. 

Two  things  affect  the  problem :  first,  the  particular  form  of 
/or  V\  and  second,  the  conditions  which  ai'e  to  hold  at  the 
limits.  With  regard  to  the  first  we  may  observe  that  there 
can  be  no  discontinuity  unless  /or  F be  of  such  a  form  that 
the  fundamental  equation/^  —  c  can  furnish  more  than  one  real 
value  of  y' .     Thus,  in  Prob.  I.,  the  fundamental  equation  is 

—  -^         =  c,  which,  because  s/  \ -^  y  is  supposed  to  remain 

positive,  can  be  satisfied  by  one  value  of  y'  only,  so  that  in  this 
case  no  discontinuity  is  possible. 

Second,  when  the  fundamental  equation  gives  several  real 
values  of  y,  and  a  combination  of  them  satisfies  all  the  other 
conditions  of  the  question,  the  necessity  for  the  employment 
of  this  combination,  or  discontinuous  solution,  generally  arises 
from  the  fact  that  the  points  to  be  joined  are  fixed.  More- 
over, as  we  in  whole  or  in  part  remove  this  restriction  from 
one  of  the  limits,  we  decrease  the  probability  that  these  val- 
ues can  be  combined  at  all ;  that  is,  that  discontinuity  will  be 
possible ;  and  even  when  it  still  occurs,  it  appears  generally 
rather  admissible  than  necessary. 

235.  When /is  a  function  oi  y"  or  y^' only,  admissible,  but 
not  necessary  discontinuity  is  still  more  likely  to  occur.  Let 
us  consider,  as  an  illustration,  a  particular  case  of  Prob.  IV. 

Problem  XXXIX. 

Let  it  be  required  to  maximize  or  minimize  the  expression 

supposing  the  limiting  values  of  x  and  y  only  to  be  fixed. 


300  CALCULUS  OF  VARIATIONS, 

Proceeding  as  usual,  we  obtain 

f'  =  §r  =  2[aY-^^=c.^  +  c^.  (2) 

But  <5>/  and  6y^  are  not  zero,  so  that  their  coefficients  //  and 
//  must  severally  vanish ;  and  assuming  the  origin  at  one  of 
the  fixed  points,  we  readily  see  that  c^  and  c^  also  vanish,  so 
that  (2)  gives 

«>"-  — =  0  (3) 


and 


y"=^±-^=±B,  (4) 


Therefore,  by  integration,  we  obtain 


^=±^  +  C,x^C„  (5) 


in  which,  because  the  origin  is  at  one  of  the  fixed  points,  C^ 
must  vanish,  and  then  C,  must  be  determined  by  making  the 
parabola  pass  through  the  second  fixed  point,  whose  co-ordi- 
nates must  satisfy  the  equation 

The  term  of  the  second  order  is 

which  is  positive  for  either  value  of  y",  thus  giving  a  mini- 
mum. 


DISCONTINUOUS  SOLUTIONS.  30 1 

We  have  here  also  the  least  value  of  U.  For  we  may 
write 

«>""  +yi=  [^y  - ^7)  +  ^'^'1'%  (7) 

h  h 

which,  by  makinpf/^  either  +  -  or >  reduces  to  2a^U'. 

•^  a  a 

236.  Here  no  discontinuous  solution  can  be  necessary, 
because  we  can  always  join  the  two  fixed  points  by  a  para- 
bolic arc,  in  which  /'  shall  be  -I —  or ;  and  also,  we  have 

a  a 

then  the  least  value  of  U.  Still,  a  discontinuous  solution  is 
always  admissible.  For  Ave  can  also  always  pass  from  the  first 
to  the  second  fixed  point  by  some  combination  of  parabolic 
arcs,  each  of  which  will  satisfy  (5),  but  will  differ  in  the  values 
of  C^  and  C^. 

Now   it   is   evident  that  all  these  arcs,  having  y"  either 

-)-  -  or ,  will  satisfy  the  equation  M  =0,  and  it  remains 

a  a 

only  to  show  that  they  will  also  make  the  terms  in  ^6^  which 

remain  without  the  integral  sign  vanish. 

Consider  two  of  these  arcs  meeting  at  the  point  x^^  y^.    The 

terms  arising  for  this  point  are 

But  since  the  equation  y  =  ±  -  holds  for  both  arcs,  f  and  -^ 

a  dx 

must  vanish  for  both,  thus  rendering  the  expression  just  given 

likewise  zero ;  and  similarly  for  any  number  of  arcs. 

Here  the  discontinuous  solution  consists  of  parabolic  arcs 

which  may  meet  at  finite  angles,  and  the  value  of  U,  and  also 

that  of  the  terms  of  the  second  order,  is  the  same  for  either 

solution. 


302  CALCULUS   OF  VARLATIONS. 

Problem  XL. 

237.  It  is  required  to  determine  the  solution  of  Prob.  XV, 
when  the  length  of  the  given  line  exceeds  that  of  the  semi-circum- 
ference described  upon  the  line  Joiniftg  the  two  fixed  points  as  a 
diameter. 

We  can  of  course  always,  by  taking  the  radius  sufficiently 
great,  join  two  points  by  a  circular  arc,  whatever  the  length 
of  that  arc  may  be  required  to  be.  But  we  cannot  here  ex- 
tend the  arc  beyond  i8o°;  because  then  there  would  be  be-, 
yond  Jo  and  y^  both  a  convex  and  a  concave  portion  of  the 
arc ;  and  besides  being  compelled  to  count  a  portion  of  the 
area  twice,  these  portions  would,  as  we  have  seen  in  Art.  95, 
give  opposite  signs  to  the  terms  of  the  second  order.  Indeed, 
whatever  may  be  the  solution,  we  would  most  naturally  un- 
derstand the  problem  to  imply  that  we  are  not  to  go  beyond 
the  production  of  the  ordinates  y^  and  j/, ;  that  is,  beyond  the 
lines  whose  equations  are  x  —  x^  and  x  =  x^,  which  may  there- 
fore be  considered  as  boundaries  which  we  must  not  trans- 
gress. 

We  would  therefore  feel  certain  that  the  solution,  at  least 
so  far  as  discoverable  by  the  calculus  of  variations,  can  consist 
only  of  what  will  satisfy  the  equation  M  —o,  with  perhaps 
some  portion  of  these  boundaries,  unless  indeed  some  other 
boundary  can  be  discovered. 

Let  us  now  see  what  can  be  obtained  in  the  usual  way. 
We  have 


u = £> + '^  ^'  +^1  '''■  =£?  f^-^-^' 


M=,-4^-JL^_,        and     X ^=_-  =  .. 


(I) 


Now  the  last  equation  will  be  satisfied  by  y'=  co  ,  because  we 
shall  then  obtain  x  —  a  =  c,  which  is  therefore  a  particular  or 


DISCONTINUOUS  SOLUTIONS.  303 

singular  solution,  being  the  equation  of  a  right  line  perpen- 
dicular to  X.  But  any  such  line  will  reduce  M  to  unity,  so 
that  we  can  only  employ  one  or  both  bo'undaries  joined  to  a 
circular  arc,  because  that  arc  gives  the  only  general  solution 
of  the  equation  M  ^=  o. 

Moreover,  we  cannot  assert  that  c  must  retain  in  this  case 
the  same  value  throughout  [/,  For  the  terms  without  the 
integral  sign  at  either  point  of  junction  of  the  arc  and  line  are 
of  the  general  form 

«i(.7r#7)r(-.^-^),H-        « 

which  in  order  to  vanish  will  require  that  y  shall  at  these 
points  mean  the  same  thing  for  the  arc  and  the  line  ;  that  is, 
that  they  shall  be  tangent.  Hence  we  are  not  confined  to  one 
boundary,  but  are  at  liberty  to  employ  both. 

238.  As  the  infinite  values  of  y  will  render  our  investiga- 
tions untrustworthy,  we  must,  in  order  to  determine  whether 
the  proposed  combination  be  the  real  solution,  transform  to 
polar  co-ordinates.  Take  the  pole  at  any  point  on  the  axis  of 
X,  between  x^  and  x^,  regarding  that  axis  as  the  initial  line,  and 
denoting  by  v  the  angle  which  any  radius  vector  r  makes  with 
this  initial.  Then  it  is  plain  that  W  must  have  the  general  form 
given  in  equation  (3),  Prob.  XX 1 1 1.,  except  that  the  limits 
will  not  be  the  same.  For  let  v^  and  z\  be  the  respective 
ciUgles  which  the  radii  r„  and  r^  drawn  to  the  two  fixed  points 
make  with  the  initial.  Then  we  need  only  consider  the  integ- 
ral from  T'o  to  -c\,  because  although  all  the  area  in  question  is 
not  comprised  within  the  limits,  still  the  two  remaining  tri- 
angles v/hich  are  included  between  the  initial  and  the  respec- 
tive radii  and  ordinates  undergo  no  variations. 

We  are,  then,  to  maximize  the  expression 


^  =X"'  {  7  +  '^  ^'-^  +  '■'"  1  -^^  =X"  ^'^^^   ' 


(3) 


304  CALCULUS  OF  VARLATIONS. 

Then,  since  Sr^  and  dr^  vanish,  if  we  suppose  U  divided  as  our 
solution  requires,  we  shall  have 

-\-fJ"M6r  dv  J^fJ'MSr  dv  ^-£'M  dr  dv,  (4) 


where 


i\/r  \  ^^  d        ar  .  ^ 

S/r'^  +  r"      dv  i/r'  +  r''  ^^^ 


Then  to  make  the  terms  without  the  integral  sign  vanish,  we 
must  have  r/—  r/  and  r/=  r/,  which  agrees  with  the  result 
from  equation  (2).  We  also  know  that  the  circular  arc  will, 
so  far  as  it  extends,  reduce  M  to  zero,  so  that  the  second  in- 
tegral in  (4)  will  vanish,  leaving  only  the  rectilinear  portions 
to  be  examined. 

Now  along  either  of  these  lines  r  cos  v  is  constant,  so  that 
by  differentiation  we  find 

,      r  sin  7^ 

r  = =  r  tan  v, 

cosz/ 


|/^2  _|_  ^/2  _  ^  |/j  _j_  ^^j^a  V  =  r  sec  V 


zo^v 


r  .  a         r 

cos  V,     =  sm  ^,    — ,  =  cos  V. 


Therefore  along  either  of  the  rectihnear  portions  M  reduces 
to  r.  But  for  these  boundaries  dr  is  always  negative,  so  that 
dC/ becomes  a  negative  quantity  of  the  first  order. 


DISCONTINUOUS  SOLUTIONS.  305 

Hence,  if  we  vary  the  whole  line,  we  are  sure  of  a  maxi- 
mum without  examining  the  terms  of  the  second  order ;  but 
if  we  vary  the  arc  only,  such  examination  would  be  necessary. 
In  this  case  we  can  again  employ  plane  co-ordinates,  and  we 
have  already  shown  that  (^t/  would  then  become  a  small  nega- 
tive quantity  of  the  second  order. 

239.  If  jKi  and  jKo  be  not  equal,  the  arc  in  a  continuous  so- 
lution cannot  equal  the  semi-circumference  having  as  its 
diameter  the  line  joining  the  fixed  points.  Let  A  and  B  be  the 
fixed  points,  and  let  y^,  the  ordinate  of  A,  be  less  than  jj,  the 
ordinate  of  B.  Let  A  Che  drawn  parallel  to  x,  C  being  upon 
the  ordinate  jj,  and  bisect  AB  at  D  by  the  perpendicular  DE, 
E  being  on  A  C. 

Then  the  limit  of  the  continuous  solution  will  be  reached 
when  the  arc  becomes  tangent  to  the  ordinate  y^ ;  that  is,  when 
its  tangent  at  A  is  perpendicular  to  AC.  Then  it  is  evident 
that  the  centre  of  the  circle  will  be  at  E.  Now  s  being  the 
length  of  the  arc,  and  R  its  radius,  we  shall  have  the  following 
equations : 

AD=\^{x,-x:f-\-^y,-y:)\ 


R  =  AD  sec  EAD  =:ABVi+  tan=  EAD, 

t^nEAD=^^i^^^-^, 

•^1  —  ^0 

Then  s  can  be  determined  by  equation  (10),  Art.  91.  De- 
note this  particular  value  of  s  by  /.  Then  if  /,  the  length  of 
the  given  line,  be  somewhat  greater  than  /,  the  line  must  be 
first  extended  along  the  ordinate  y^,  produced  a  certain  dis- 
tance l\  until  a  point  is  reached  at  which  the  same  construc- 
tion can  be  made  as  at  ^.  Then  all  the  equations  just  given 
will  be  rendered  true  by  merely  substituting  for  y^,  y^  -{- 1',  so 


3C6  CALCULUS   OF    VARIATLONS. 

that  the  new  values  of  R  and  s  may  be  found  in  terms  of 
x^,  jKo,  -^1,  y^  and  I' ,  and  then  we  have  the  additional  equation 
l—l'^s,  so  that,  /  being  given,  V  can  be  also  determined. 
This  construction  will  hold  until 

^=/i->'o+-(-^i-^o), 

when  the  arc  will  become  a  semi-circumference.  If  then  /  be 
still  further  increased,  we  must  retain  the  same  semi-circum- 
ference, but  also  produce  y^  as  well  as  y^  a  certain  distance  I' , 
Then  we  shall  have 

Hence,  as  /  is  supposed  to  be  given,  I'  will  be  determined, 
and  this  construction  will  hold  when  /  is  indefinitely  ex- 
tended. 

We  must,  in  closing,  call  attention  to  the  fact  that  this 
problem,  when  discussed  by  plane  co-ordinates  as  at  the  be- 
ginning, affords  another  instance  to  show  that  necessary  dis- 
continuous solutions  are  generally  suggested  by  the  funda- 
mental equation,  even  when  they  do  not  satisfy  at  all  the 
equation  M  —o. 

Problem  XLI. 

24-0.  It  is  required  to  determine  the  discontinuous  solution  i?t 
Prob.  XIX. 

It  will  be  remembered  that  when  x\  is  zero,  x^  becomes  a 
definite  function  of  the  given  volume,  so  that  if  we  require 
the  second  point  on  the  axis  of  x  to  be  fixed — that  is,  x^  to 
have  a  given  value — then,  unless  that  value  happen  to  satisfy 

the  equation  x^  —  V  -—-,  where  tj  is  the  volume,  we  must 
resort  to  some  discontinuous  solution,  if  any  solution  be  pos- 


DISCONTINUOUS  SOLUTIONS.  307 

sible.  (See  equation  (8),  Prob.  XIX.,  observing  that  c  there 
was  shown  to  equal  x^^ 

Now  as  ^^in  this  problem  does  not  admit  of  the  usual 
transformation,  because  it  contains  no  variation  but  that  of  j/, 
the  fundamental  equation  is  found  by  equating  to  zero  the  co- 
efficient of  6y  dx  in  equation  (2)  of  that  problem,  which  gives 
either  J/  =  o,  or  else  equation  (3). 

This  suggests  that  if  the  value  of  x^  be  too  great — that  is, 


greater  than  j/  -—  —  the  solution  will  consist  of  a  curve  satisfy- 

ing  equation  (4),  and  extending  from  the  origin  to  some  point 
x^  on  the  axis  of  ;r,  x^  being  less  than  ,r,,  and  then  of  the  axis 
itself  from  x^  to  x^ ;  and  that  if  x^  be  too  small,  the  solution 
may  consist  of  the  same  solid  extended  to  x„  beyond  x^,  and 
then  of  the  axis  from  x^  to  x^,  the  solutions  thus  being  similar 
to  those  in  the  case  of  the  sphere. 

Now  the  terms  of  the  second  order,  as  we  see  from  equa- 
tion (2),  are 

dU=  r^  \  a-\-x-^^^-^'-  1   dfdx. 
But  if  we  put  7  =  0,  and  for  a  its  value  -^-»  we  shall  obtain 


6U^.S 


\-^-\-—}i^fdx, 

{    2C^  2X^  ) 


where  the  integral  extends  over  the  rectilinear  portion  only; 
while  if  we  vary  the  generating  curve,  (5^6^  will  take  the  form 
given  in  equation  (11),  where  the  integral  will  extend  from 
x^  to  x^,  and  will  be  negative  whether  x„  be  less  or  greater  than 
x^.  Hence,  observing  that  c  —  x^,  the  entire  variation  may  be 
written 


308  CALCULUS  OF  VARIATIONS. 

Now  in  order  that  U  may  be  a  maximum,  the  second  integral 
in  (i)  must  also  become  negative,  otherwise  the  sign  of  dlJ 
would  become  ambiguous.  But  any  element  of  this  integral 
will  evidently  become  negative  or  positive  according  as  x^  is 
less  or  greater  than  x.  Now  when  the  solid  does  not  extend 
to  the  second  fixed  point,  x  for  the  rectilinear  part  is  greater 
than  x^,  and  the  same  will  be  true  when  the  solid  extends  be- 
yond the  second  fixed  point,  provided  we  agree,  as  explained 
in  Art.  195,  to  regard  x  for  the  rectilinear  part  as  still  increas- 
ing from  x^  to  x^ ;  so  that  under  this  supposition  we  have 
always  a  maximum. 

241.  But  the  solution  in  the  case  in  which  the  solid  ex- 
tends beyond  the  second  fixed  point  may  not,  perhaps,  be 
deemed  altogether  satisfactory.  For  in  the  volume  which  is 
generated  by  the  derived  curve,  we  are  obliged,  as  before,  in 
the  case  of  the  sphere,  to  reckon  twice  that  generated  by  ^y 
along  the  rectilinear  part,  and  also  to  regard  its  attractive 
force,  when  counted  the  second  time,  as  what  it  would  be  if 
each  element  were  placed  as  far  beyond  x^  as  it  now  falls  short 
of  that  point. 

We  do  not,  therefore,  in  reality,  compare  the  attraction  ex- 
erted by  the  primitive  solid  with  that  which  would  really  be 
exerted  by  the  derived  solid,  but  merely  with  what  the  attrac- 
tion of  that  solid  would  be  if  the  attraction  of  any  particle 
could  vary  inversely  as  the  square  of  the  estimated  value  of 
X,  instead  of  its  actual  value. 

Thus  we  have  here  merely  a  sort  of  theoretical  or  imagi- 
nary solution,  not  properly  capable  of  geometrical  representa- 
tion, and  presenting  itself  possibly  somewhat  as  do  imaginary 
roots  in  the  theor}^  of  ordinary  equations.  But  the  condition 
that  the  solid  is  to  meet  the  axis  of  ;ir  at  a  second  fixed  point 
may,  as  Prof.  Todhunter  has  suggested,  be  more  naturally 
understood  to  mean  that  the  solid  is  not  to  stretch  beyond  the 
line  whose  equation  is  ^  =  x^.     Then  c  in  (4)  would  no  longer 


DISCONTINUOUS  SOLUTIONS  3O9 

be  equal  tO-Tj,  but  could  be  determined  from  equation  (7)  by 
making  the  limits  o  and  x^,  since  x^  and  v  are  both  given ;  and 
then  all  the  conditions  of  the  question  could  be  fulfilled. 

But  should  neither  of  these  solutions  prove  satisfactory, 
we  are  still  at  liberty  to  suppose  that  there  is  no  solution, 
since  it  is  evidently  possible  to  assume  such  conditions  in  any 
problem  as  will  render  any  solution  either  continuous  or  dis- 
continuous impossible ;  as,  for  example,  if  in  Prob.  XV.  we 
should  assume  the  given  line  to  be  shorter  than  the  right  line 
joining  the  two  fixed  points. 


Problem  XLII. 

242,  It  is  required  to  discover  the  nature  of  the  discontinuous 
solution  in  Prob.  XXI 

Here,  as  will  appear  from  reference  to  the  problem,  the 
continuous  solution  consists  of  an  oblate  spheroid  whose  major 
axis  is  to  the  minor  as  ^2  is  to  i  ;  that  is,  whose  eccentricity 

is  — =r,  ^^  the  square  of  the  semi-minor  axis,  being  equal  to  x^. 

V2 

Hence,  if  the  given  volume  be  greater  or  less  than — ,  the 

solution,  if  any  exist,  must  be  discontinuous. 

But  the  fundamental  equation  in  this  case,  as  will  be  seen 
from  equation  (4),  is 

y  {/  -f  2.r'  —  2d')  =  o, 

which  gives  either  jj/  =  o  or  equation  (5),  which  is  the  equa- 
tion of  the  generating  ellipse.     Let  A  and  B  be  the  two  fixed 


3IO  CALCULUS  OF  VARIATIONS. 

points  on  the  axis  of  ;r,  and  C  the  origin,  which,  it  will  be 
remembered,  was  required  to  be  midway  between  A  and  B. 
Then  it  is  suggested  that  the  discontinuous  solution  might  be 
that  represented  in  the  figure,  where  the  generating  ellipse  is 
DE  or  FG,  according  as  the  given  volume  is  less  or  greater 

than -. 

3 
Here,  then,  U  for  either  case  may  be  divided  into  three  in- 
tegrals, extending  respectively  from  x^  to  x^,  from  x^  to  x^, 
and  from  x^  to  x^ ;  x^  being  in  the  first  case  the  abscissa  of  D, 
and  in  the  second  that  of  i%  and  x^  being  that  of  E  or  G.  We 
must  also  recollect  that  in  the  second  case  x^  and  x^  are  thus 
estimated : 

X,  =  -  (CF+  FA)         and    x,  =  CG+  GB. 

Now  we  have  seen  (Art:  120)  that  the  terms  of  the  second 
order  for  the  ellipse  reduce  to    /  ysydx,  and  to  obtain  the 

variation  of  the  rectilinear  portions  we  have  merely  to  make 
f  zero  in  the  first  equation  of  that  article,  so  that  we  have 

d[/  =fj\^"  -  a')6/dx  +  fJ'/S/dx  +  fj\x'  -  a'')6/dx. 

To  render  the  first  and  third  integrals  positive,  we  must  have 
x""  >  d^;  and  since  d"  =  x^  =  x^,  if  we  estimate  x  for  the  recti- 
linear part  as  already  explained,  this  condition  will  be  fulfilled 
in  either  case,  and  ^becomes  a  minimum. 

But  since  the  solids  generated  by  both  the  primitive  and 
the  derived  curve  are  to  be  revolved  about  the  axis  of  j/,  it 
must  appear  that  when  the  sohd  extends  beyond  A  and  B  the 
solution,  like  that  of  the  preceding  problem,  is  merely  theo- 
retical or  imaginary.  These  problems  also  resemble  each 
other,  and  differ  from  all  others  which  we  have  considered, 
in  that,  as  t/ contains  x  and  7  only,  there  are  no  terms  m  6U 


DISCONTINUOUS  SOLUTIONS.  3II 

without  the  integral  sign ;  and  hence  the  equation  L  =  0 
gives,  without  integration,  the  equation  of  the  required 
curve,  and  there  are  no  terms  to  consider  at  the  limits. 


Problem  XLIIL 

243.  //  is  required  to  determifte  what  discontinuous  solutions 
can  present  themselves  in  the  discussion  of  Prob.  XX. 

Here  the  continuous  solution  is  an  hypocycloid,  in  which 
the  radius  of  the  rolling  circle  is  one  third  that  of  the  fixed 
circle.  But,  by  the  closing  remarks  of  Art.  1 16,  it  appears  that 
this  solution  cannot  hold  when  the  given  volume  becomes  less 

than ^,  where  b  is  the  radius  of  the  given  base ;  so  that  if 

the  given  volume  be  less  than  this  quantity,  the  solution,  if 
there  be  any,  must  be  discontinuous. 


Let  AD  be  the  axis  of  x,  and  DB  the  radius  of  the  gen- 
erating base.  Then,  since  the  volume  was  to  be  upon  the 
given  base,  we  would  naturally  infer  that  when  the  volume 
becomes  too  small,  the  generating  curve  would  consist  of  an 
arc  AC  oi  an  hypocycloid,  and  a  portion  CB  of  the  radius  of 
the  base.  In  fact,  we  may  understand  the  conditions  of  the 
problem  to  imply  that  the  solid  is  always  to  be  upon  a  portion 
of  the  base. 

244.  Before  considering  whether  this  solution  is  also  sug- 
gested by  the  calculus  of  variations,  we  will  show  that  it  is  in 
some  cases  the  solution  required. 


312  CALCULUS  OF  VARLATLONS. 

For  the  solid  generated  by  ^(7  the  resistance  will  evidently 
be  27tj^  "Y  /2  ^^^  ^^<^  fo^"  the  ring  generated  by  CB  it  will 
be  7t{b'^  ~ yi)'     Hence  we  may  minimize  the  expression 

where  we  are  to  regard  jj,  the  ordinate  of  C,  as  variable,  but 
the  other  terms  at  the  limits  as  fixed.  Now  taking  the  varia- 
tion of  U  under  this  supposition,  transforming  it  as  usual,  and 
making  M  vanish,  we  shall  obtain,  as  in  Prob.  XX.,  equation 
(4),  which  will  be  of  course  the  differential  equation  of  the 
hypocycloidal  arc  AC.     But  we  have,  after  malting  J/ vanish, 

and  to  satisfy  this  equation,  we  must,  since  y^  is  not  zero,  have 

+  (I  -^ry'J      °' 

which  gives  y  =  ±  i'  Thus  it  appears  that  the  generating 
curve  must  meet  the  ordinate  of  B  at  an  angle  of  45"". 

246.  To  determine  the  sign  of  the  terms  of  the  second 
order,  we  must  observe  that  the  terms  under  the  integral  sign 
in  the  value  of  ^ given  in  (i)  equal  2^  in  Prob.  XX.  Hence 
we  shall  obtain  from  the  variation  of  these  terms  twice  the 
second  member  of  equation  (19),  Art.  117.  But  we  shall  also 
obtain  from  this  integral  a  term  without  the  sign  of  integra- 
tion. For  (19)  was  obtained  under  the  supposition  that  6j/, 
and  (^Ko  vanish.  When,  however,  this  is  not  the  case,  we  must, 
as  we  see  from  equation  (6),  Prob.  VHI.,  add  to  the  second 
member  of  (19)  the  terms 

J  (//<5j," -•/.%.").  (2) 


DISCONTINUOUS  SOLUTIONS.  3I3 

which  will  give  in  this  case  the  additional  term 

HI  +/  )  '  1 

and  as  this  is  cancelled  by  the  term  of  the  second  order  arising 
from  the  variation  of  —  y^  in  U,  SU  becomes  merely  twice 
the  second  member  of  (19),  which  is  positive. 

246.  Thus  we  have  a  minimum  if  c  have  any  real  value. 

Now  because  j/  =  ±  i,  taking  the  positive  sign,  we  have,  from 

equation  (4),  Prob.  XX.,  which,  it  will  be  remembered,  is  the 

c 
fundamental  equation  in  this  case  also,  jj/j  —  —  ;  and  it  is  also 

4 
shown  by  operations  of  the  differential  and  integral  calculus 
only,  that  the  given  volume,  v',  will  in  this  case  be 

v'  =  -^ (3) 

1920  ^^'' 

Hence,  when  v^  is  given,  c  and  j/j  are  at  once  determined. 

Now  v'  can  be  given  as  small  as  we  please,  but  it  cannot 
be  as  great  as  we  please.  For  j\  must  not  exceed  d ;  and  as 
c  =  4jj,  v^  evidently  increases  as  we  increase  y^,  and  must  have 
its  greatest  value  when  ji  =  b — that  is,  when  c  :=4d — and  then 
(3)  gives 

.-135^.  (4) 

30  ^^^ 

We  see,  then,  that  if  the  given  volume  be  less  than -,  we 

must   always   employ   the   discontinuous   solution ;    if   it   be 

s^reater  than  — — ,  we  must  alwavs  employ  the  continuous 

30  .  ^     J 

solution ;  but  if  it  He  between  these  values,  then  we  shall  have 
two  mmima,  and  must  determine  which  will  give  the  smaller 


314  CALCULUS   OF  VARIATIONS. 

resistance.  This  determination  must,  however,  as  in  former 
cases,  be  effected  by  the  ordinary  calculus  alone,  using,  of 
course,  any  equation  which  has  been  thus  far  obtained. 

It  will  be  sufficient  here  to  give  the  necessary  formulse 
and  results.  Let  2\  denote  the  angle  whose  tangent  is  jk/. 
Then,  R  being  the  resistance.  Prof.  Todhunter  shows,  by 
methods  of  the  ordinary  calculus,  that  for  the  continuous 
solution 


„  __     nb^     (i  sin^  v. 


^-^^i.  (5) 


cos  v^  \      4 
and  that  for  the  discontinuous  solution 

where,  since  v'  is  supposed  to  be  a  given  quantity,  v^  can  be 
determined  from  equation  (12),  Art.  116,  and  c  from  equation 
(3)  of  this  article.  Now  if  we  take  the  extreme  values  of  v', 
for  which  two  solutions  are  possible, 

^'  — ±         and     v'  =  -^ ,  (7) 

5  30  ^^^ 

we  shall  find  that  the  two  solutions  coincide  for  the  first,  R 

being  in  either  case  ^- — -,  and  for  the  second  value  of  v'  we 
20 

shall  find  that  R  is  less  for  the  discontinuous  than  for  the  con- 
tinuous solution.     For  we  have  in  the  first  case 


I  _  ^ — £_  K  —  Ttb^  X. 440 1 2,  nearly  ; 


and  in  the  second 


_  9^^' 


20 


DISCON-TINUOUS  SOLUTIONS.  31 5 

It  is  also  shown,  by  determining  the  sign  of  -— ,  that  both  for 

dv 

the  continuous  and  discontinuous  solution  R  decreases  as  v' 

increases.     Hence,  from  what  has  been  already  shown,  it  will 

appear  that,  when  there  are  two  solutions,  the  discontinuous  is 

that  which  will  always  give  the  smaller  resistance. 

247.  It  will  be  remembered  that  in  Prob.  XX.  we  con- 
sidered only  the  case  in  which  v  is  supposed  to  be  zero  when 

y  is  zero.    But  if  we  supposed  that  when  y  is  zero  v  is  — ,  and 

measure  the  arc  s  from  that  point,  then  we  shall  have,  from 

equation  (lo)  of  that  problem,  ^  = cos  yu. 

Here,  on  account  of  the  infinite  value  of  y\  our  investiga- 
tion of  the  terms  of  the  second  order  will  not  be  satisfactory, 
and  we  will  therefore  adopt  y  as  the  independent  variable. 
Then  C^  becomes 

Hence,  to  the  second  order, 

Therefore,  by  making  the  terms  of  the  first  order  vanish, 

2yx' 
2ay'^  —  - — Y^ — —  =  a  constant,  which  must  be  o ; 

(I  -j-  ;r  ) 

and  this  must,  of  course,  lead  to  the  hypocycloid,  as  before. 
Then,  as  the  terms  of  the  first  order  vanish,  we  shall  have 


'^^=r>(S^<^-'^^-^' 


3l6  CALCULUS   OF  VARLATLONS. 

which  is  negative  so  long  as  x'"^  does  not  exceed  -  ;  that  is,  so 

7t 

long  as  ^1  is  not  less  than  -      Thus  in  this  case  the  resistance 

becomes  a  maximum,  provided  we  can  determine  real  values 
for  c. 

Now,  as  before,  it  is  shown  that  in  this  case 

cos^  v^      1 3  cos^  2^       1 5  cos*  v^      7  cos'^  ^1  _|_  i 

~3  10        '  8  6        ^3   ,    . 

V  =  7tc  COS  V,      -^  I  ,  -^ .  ( 10) 

Also,  because  equation  (5),  Prob.  XX.,  holds,  we  shall  find 
that  here,  as  in  equation  (13)  of  the  same  problem, 

^  =  sin' z/j  cos  2/j,  (11) 

and  from  these  two  equations  c  must  be  determined.  It  is 
evident  that  v'  can  be  made  as  small  as  we  please ;  but  it  can- 
not be  taken  as  great  as  we  please,  because  it  decreases  with 
v^ ;  and  in  order  to  have  a  maximum,  v^  must  not  be  less  than 


(12) 

which  is  therefore  the  greatest  admissible  value  of  v' . 

248.  We  are  naturally  led  to  inquire  whether  there  will 
be  any  discontinuous  solution  when  v'  exceeds  the  value  just 
given. 

Since  the  solid  is  to  be  bounded  by  the  given  base,  the 
only  suggestion  which  presents  itself  is  that  y^  may  now  be 
greater  than  b.  Then,  when  y  is  the  independent  variable,  U 
will  have  the  form  given  in  (8).  But  now,  as  7,  is  variable, 
we  must,  when  we  vary  U,  increase  also  the  limit  y^  by  dy^ ; 


7t 

3* 

But  when  v^  =  -,  we  shall  find 

,          32171^ 

I215 

DISCONTINUOUS  SOLUTIONS.  317 

that  is,  we  must  add  to  the  terms  of  the  first  order  in  (9)  the 
term 

V,  dy,     or      |  YjIV'  +  ^^^'^'  \  ^^'' 

Now  the  coefficient  of  Sx^  will  necessarily  vanish,  but  we  can- 
not also  make  V^  vanish.  Hence  SU  \.o  the  first  order  will 
not  vanish  ;  and  as  dy^  may  have  either  sign,  U  will  be  neither 
a  maximum  nor  a  minimum. 

249.  A  somewhat  curious  point  is  here  noticed  by  Prof. 
Todhunter,  which  it  may  be  useful  to  consider. 

Let  A  be  the  distance  of  the  base  from  the  origin.  Then 
we  may  evidently  consider  the  solid  as  composed  of  cylindri- 
cal shells  whose  radius  is  y,  thickness  dy^  and  length  A  —  x. 
Then,  instead  of 

£yfdx       or      fjy/^'dy, 

the  volume  may  be  written  /   ^27ry{A  —  x)dy.    Therefore  with 

this  value  of  v'  we  are  to  maximize  or  minimize  the  expres- 
sion 

+J2'\-"+'^jj^f\""l>-  (.3) 

Hence,  by  integration,  we  obtain 

—  a/-\-     ^         —  a  constant,  which  must  be  o. 


3l8  CALCULUS   OF  VARIATIONS. 

This  equation  is  in  reality  the  same  as  that  which  we  ob- 
tained before,  and  leads,  therefore,  to  the  hypocycloid.  Thus 
the  integral  in  (13)  will  vanish,  and  so  also  will  the  terms  at 
the  lower  limit,  because  there  y  is  zero ;  but  the  terms  at  the 
upper  limit  will  not  vanish,  so  that  we  have,  by  the  last  equa- 
tion. 

Now  since  the  base  is  a  boundary  which  we  may  not  pass, 
dx^  is  essentially  negative,  and  thus  ^U  becomes  a  positive 
quantity  of  the  first  order,  indicating  that  we  have  a  condi- 
tioned minimum,  which  result  would  seem  to  show  that  we 
can  never  have  a  solid  of  maximum  resistance,  thus  conflict- 
ing with  what  has  been  before  proved. 

250.  To  explain  this  difficulty,  let  AB  be  the  primitive 
curve,  and  suppose  we  wished  to  pass  to  a  derived  boundary 
EDB,  where  DB  is  parallel  to  x,  and  infinitesimal. 


Then  we  could  not  derive  this  boundary  from  AB  by  inhni- 
tesimal  changes  in  y  and  y,  although  we  could  by  such 
changes  in  x  and  x'. 

This  assertion,  which  Prof.  Todhunter  takes  no  pains  to 
establish,  may  at  first  appear  incorrect,  because  we  seem  to 
have  given  x^  a  finite  variation  in  order  to  obtain  DB,  which 
would  be  inadmissible.  But  the  position  appears  to  be  sound, 
since  we  should  regard  x^\  after  being  varied,  not  as  the  tan- 
gent of  the  inclination  of  DB  to  y,  but  as  that  of  the  inclina- 
tion to  y  of  the  tangent  to  the  derived  curve  at  D,  supposing 


DISCONTINUOUS  SOLUTIONS.  319 

the  curve  ED  produced  beyond  D.  Then  Sx^  need  not  be 
finite. 

Now  since  y^  is  fixed,  we  shall  (unless  in  the  last  article  we 
make  dx^  zero,  in  which  case  all  the  terms  of  the  first  order 
will  vanish,  and  there  will  be  no  difficulty)  be  obliged  to  pass 
to  a  derived  curve  terminated  by  DB,  DB  being  numerically 
equal  to  Sx^.    Still,  so  long  as  we  adopt  for  the  volume,  as  we 

did  in  (8),  the  expression  nj    y'x'dy,  we  cannot  pass  to  such 

a  boundary  as  we  have  been  considering ;  because  although 
the  expression  just  given  will  represent  the  volume  generated 
by  the  primitive,  still,  when  we  change  x'  into  x'  -\-  dx\  and 

write  v'  =  TtJ  y{x'  -\-  dx')dy,  v'  can  only  represent  the  vol- 
ume generated  by  ED,  neglecting  entirely  that  generated 
hy  DB. 

Hence  we  conclude  that  the  form  of  v'  adopted  in  (8)  is 
not  general  enough  to  permit  of  a  full  discussion,  as  it  will 
not  allow  every  change  in  the  form  of  the  solid  which  the 
calculus  of  variations  would  in  this  case  sanction.  We  see, 
also,  that  we  can  have  a  solid  of  maximum  resistance  only 
under  the  condition  that  jKi,  the  radius  of  the  generating  base, 
shall  be  invariable,  and  that  the  curved  part  of  the  solid  shall 
always  extend  to  the  circumference  of  the  base. 

251.  We  have  in  this  discussion  a  remarkable  confirmation 
of  the  principle  often  before  stated — that  when  by  variations 
we  have  obtained  conditions  which  render  any  definite  inte- 
gral U  a  maximum  or  a  minimum,  we  are  not  necessarily  war- 
ranted in  asserting  more  than  that  C/  is  a  maximum  or  a  mini- 
mum with  respect  to  admissible  variations.  For  the  sohd  of 
minimum  resistance  which  we  obtained  in  Prob.  XX.  is  not 
the  solid  of  least  resistance,  since  by  taking  a  zigzag  boundary 
it  could  be  still  further  diminished,  although  we  could  not 
pass  to  such  a  boundary  by  the  calculus  of  v? nations.  More- 
over, our  solid  of  maximum  resistance  is  such  so  long  only  as 


320  CALCULUS  OF  VARIATIONS. 

we  do  not  make  suck  a  change  in  the  form  of  the  sohd  as  in 
Art.  240.  But  a  solid  of  still  greater  resistance  would  evi- 
dently be  obtained  by  passing  to  a  boundary  in  which  y'  is 
alternately  zero  and  infinity,  although  such  a  change  of  form 
cannot  be  effected  by  the  calculus  of  variations. 

252.  It  will  be  remembered  that  in  Art.  243  we  were  led 
to  the  discontinuous  solution,  which  we  subsequently  verified, 
by  the  consideration  that  the  given  base  constituted  a  bound- 
ary, and  that  therefore  it  would  probably  form  some  portion 
of  the  solution. 

Now  we  have  found  hitherto  the  boundaries  to  be  also  in 
some  manner  suggested  by  the  fundamental  equation  which 
IS  usually  the  first  integral  of  the  equation  M  —  o,  even  when 
these  boundaries  do  not  in  reality  cause  M  to  vanish  at  all. 
In  the  present  case,  however,  the  discontinuous  solution  does 
not  appear  to  be  very  clearly  suggested  by  the  calculus  of 
variations  alone,  unless,  indeed,  U  can  be  put  under  some 
form  different  from  those  which  we  have  yet  examined.  For, 
adopting  in  succession  x  and  y  as  the  independent  variable, 
the  first  integral  of  the  equation  J/  =  o  will  be  in  each  case 
the  most  general  form  of.  the  fundamental  equation,  and  we 
shall  have 


yy 


^y  —  7     I     /2^2  =  a  constant 
(i  -\-y'J 


and 


y^' 


^^  ~  {i-\-x''f  "^  ^  constant. 


which  constant  must,  in  either  case,  be  zero,  because  the  curve 
is  to  meet  the  axis  of  x.  Therefore,  rejecting  the  solution 
J/  —  o,  we  have 

=        /'        ^     //'  H         -         ""'         -       ^'^' 

and  these  equations  lead  to  the  same  solution. 


DISCONTINUOUS  SOLUTIONS.  32 1 

Now  y  =  CO  or  x'  =  o  are  not  solutions  of  these  equations, 
unless,  indeed,  we  could  suppose  <;  ==  co  and  C  —  co .  But 
these  constants  will  not  be  infinite  for  the  curve ;  and  since 
they  are  in  each  case  the  reciprocal  of  a,  if  we  remember  that 
even  in  a  discontinuous  solution  the  constant  introduced  by 
Euler's  method  cannot,  like  a  constant  of  integration,  have 
two  values,  it  will  appear  that  c  and  C  cannot  become  infinite 
at  all. 

253.  There  would  seem  to  be  nothing  surprising  in  the 
fact  that  the  fundamental  equation  does  not  always  suggest  a 
boundary  which  does  not  cause  M  to  vanish  at  all,  and  indeed 
it  would  appear  more  remarkable  that  such  boundaries  are  so 
frequently  suggested.  Cases,  however,  like  the  present  ap- 
pear to  be  rare,  and  we  have  now  had  abundant  proof  that 
the  calculus  of  variations  does  usually  suggest  solutions  when 
they  are  possible,  and  even  when  such  suggestions  would  not 
naturally  be  expected. 

Moreover,  in  discontinuous  solutions  it  very  often  happens 
that  a  trial  solution  is  easily  reached  without  the  aid  of  varia- 
tions, or  at  least  without  examining  the  form  of  M;  and  then 
the  calculus  of  variations  affords  us  the  means  of  verifying  or 
falsifying  this  proposed  solution,  and  that,  too,  very  frequently 
without  an  appeal  to  the  terms  of  the  second  order. 

254-.  The  subject  of  the  present  section  has  been  most 
elaborately  treated  in  the  Adams  Essay,  or  Researches  in  the 
Calculus  of  Variations,  published  by  Prof.  Todhunter  in  1871, 
and  to  his  labors  its  present  degree  of  perfection  is  chiefiy 
due.  In  this  section,  which  is  little  more  than  a  condensed 
view  of  that  treatise,  we  have  endeavored  to  present  all  the 
leadmg  points  of  that  work,  and  particularly  those  points 
which  were  new  to  our  science.  All  the  examples,  there- 
fore, of  this  section  have,  with  slight  modifications,  been  taken 
from  this  essay,  although  we  have  in  no  respect  followed  its 


322  CALCULUS   OF  VARLATIONS. 

order.     We  therefore  earnestly  recommend  the  work  to  all 
who  wish  to  become  fully  acquainted  with  this  subject. 

We  have,  with  the  exception  of  Prob.  XXXIII. ,  consid- 
ered those  cases  only  in  which  the  discontinuity  ma}^  be  sup- 
posed to  arise  from  conditions  unconsciously  imposed,  or  at 
least  imposed  without  seeking  to  produce  it ;  because  it  is  only 
when  it  thus  presents  itself  that  its  origin  can  be  a  source  of 
difficulty.  It  must,  however,  be  evident  that  even  when  no 
discontinuity  would  naturally  occur  in  a  problem,  we  can 
easily  impose  such  restrictions  as  will  render  a  discontinuous 
solution  necessary,  and  the  work  to  which  we  have  referred 
is  occupied  largel}^  with  such  examples,  some  of  which  exhibit 
much  ingenuity.  But  as  these  examples,  while  affording  ex- 
cellent practice  in  this  department  of  analysis,  present  nothing 
which  has  not  been  already  considered,  it  will  be  sufficient  in 
closing  to  subjoin  one,  which  is  all  that  our  space  will  permit. 


Problem  XLIV. 

255.  It  is  required  to  find  the  path  of  quickest  descent  of  a 
particle  from  a  fixed  point  A  to  a  second  fixed  point  B^  under  the 
condition  that  the  path  is  not  to  pass  without  a  given  circular  arc 
AB,  which  is  not  to  exceed  a  quadrant ;  the  particle  starting  from 
a  state  of  rest  at  A ,  and  B  beijtg  the  lowest  point  of  the  arc. 

Assume  the  horizontal  as  the  axis  of  x.  Then,  as  in  Case 
2,  Prob.  II.,  we  shall  have 


^^0  Af7.  t/.Tn  ' 


6U=:P6y-\-fMdydx, 


(I) 


DISCONTINUOUS  SOLUTIONS.  323 

where  the  limits  and  suffixes  are  for  the  present  omitted,  and 


p^dV 


dy  ^yix^y"^ 


(2) 


ax  2j'i  ax 


where  iV^==  -— .     Now  wherever  the  sig^n  of  ^y  is  unrestricted, 

ay 

J/ must  vanish,  and  this  will  lead  to  a  cycloid  having-  its  cusps 
on  the  horizontal  line  through  A,  and  its  vertex  downward. 

But  the  cycloid  alone  can  never  be  the  solution,  because 
its  tangent  at  A  being  perpendicular  to  x,  it  is  initially  with- 
out the  circle.  Since,  then,  the  circle  is  the  only  boundary 
along  which  the  sign  of  Sy  can  be  fettered,  the  solution  must 
consist  either  of  the  given  circular  arc  alone,  or  of,  first,  a 
portion  of  that  arc  beginning  at  A,  follow^ed  by  some  combina- 
tion of  portions  of  that  arc  and  cycloidal  arcs  given  by  the 
equation  J/ =  o. 

256.  Let  the  initial  and  the  first  cycloidal  arc  meet  at  the 
point  x^,  y^.  Then  there  will  evidently  arise  in  dJJ.,  as  given 
in  (i),  the  terms  [P^  —  P^Sy^,  2iXidi  this  must  either  become 
positive  or  vanish  ;  that  is,  since  Sy^  must  be  negative,  P^  —  P^ 
must  be  negative  or  vanish.  But  if  it  were  negative,  we 
would,  as  appears  from  (2),  have  y^'  >y^',  which  would  re- 
quire the  cycloid  at  that  point  to  pass  without  the  circle, 
which  is  inadmissible.  Hence  the  coefficient  of  Sy^  being 
zero,  we  have  yj  =^yjj  and  the  circle  and  cycloid  must  be 
tangent  at  the  point  x^,  y^.  In  like  manner  they  would  evi- 
dently be  tangent  if  they  could  meet  at  any  other  point. 

257.  Let  AC  be  the  horizontal  through  A,  O  the  centre 


324 


CALCULUS  OF  VARIATIONS. 


of  the  given  circular  arc,  and  r  its  radius,  R  being  the  point 
x^yv  ^o  that  RTisdi  normal  to  the  cycloid. 


Now  take  any  point  on  ^(7  as  the  origin.     Then  the  equa- 
tion of  the  circular  arc  is 


{x-cr+{y  +  bY  =  ,^, 


(3) 


where  b  =  OC,  and  c  is  the  abscissa  of  O.     Therefore,  for  the 
circle,  we  find 


f=-- 


X  —  c 


y+b' 
Hence,  from  (2),  we  have 

N 


Vi  +/'  = 


y-^b 


(4) 


~  r 


X — c 


2j/^{j/  -\-  b)       27y\y  -f-  b) 


r  Vy 


dP_  -2y+y(x-c)  _  -2y(y  +  b)-{x-cy 
dx  ~  2ry^  ~~  2ry^{y  +  b) 

Whence,  putting  in  N  the  value  of  r'  from  (3),  we  have 

y  —  b 


(5) 


J/: 


2ry^ 


(6) 


Now  this  value  of  AI  must  either  vanish  or  become  negative 
in  order  that  /  MSydx  may  be  positive  along  the  circle,  since 


DISCONTINUOUS  SOLUTIONS.  325 

M  will  vanish  along  the  cycloid,  and  this  requires  merely  that 
y  —  b  shall  not  become  positive. 

258,  Let  /  and  a  denote  respectively  the  angles  OTC  and 
OAC.     Then 

RT=OR-OT=r--J-  =  r-  ^^^^- ;  (7) 

sm  t  sm  ^  ^'  ^ 

and  because  i^T'is  a  normal  to  the  cycloid,  we  have,  D  being 
the  diameter  of  the  generating  circle, 

^      RT  sin  t  —  sin  a  ,  ^ 

D  =  -^-  =  r ^^ .  (8) 

sm/  sm  /  ^  ^ 

If  the  cycloid  can  meet  the  circle  again  at  some  other 
point  S,  we  shall  obtain  a  similar  expression  for  D,  only  t 
will  then  denote  the  angle  which  6^5  would  make  with  ^(7, 
and  these  expressions  would  be  equal.     Hence,  regarding  / 

as  variable,  and  writing:  v  = ^- ,  we  must  be  able  to 

^  sm'/ 

effect  that  v  shall  twice  have  an  assigned  value,  or  else  the 

circle  and  the  cycloid  cannot  meet  more  than  once. 

Now  we  find 

dv                2  sin  a  —  sin  /  ,     ^ 

—  =  cos  / r-3- .  (10) 

That  is,  to  render  v  a  maximum  or  a  minimum  we  must  have 
either 

2  sin  <?:  —  sin  /  =  o        or     cos  t  =  o.  (i  i) 

Since   sin  t  cannot   exceed  unity,  if  2  sin  a  be  greater  than 

unity,  the  first  equation  cannot  be  satisfied,  and  v  continually 

increases  as  /  passes  from  a  to  -,  and  therefore  cannot  twice 

have  the  same  value ;  and  the  same  would  be  true  should  2  sin  a 
equal  unity. 


326  CALCULUS   OF  VARLATLONS. 

Neither  can  we  in  this  case  make  the  cycloidal  arc  meet 
the  circle  at  R  and  also  pass  through  B.     For 

CB=  OB  -  OC=r-  OC=:^r-b  =  r{i-  sin  a), 

and  D,  as  appears  from  (8),  must  be  less  than  CB  so  long  as  / 

is  less  than  — ;  that  is,  sin  /  less  than  unity ;  and  hence  in  this 

case  we  must  use  the  circle  alone. 

Now  since  y  -^  b  —  ER  =  r  sin  t,  and  b  =  OC  —  r  sin  a,  we 
have  y  —  b  ^=  r  (sin  t  —  2  sin  a),  which  is  in  this  case  negative, 
thus  rendering  ^f/ positive  for  the  whole  circular  arc.  But 
sin  a  =  cos  A  OB,  so  that  2  sin  <^  will  be  unity  when  AB  is 
an  arc  of  60°.  We  conclude,  therefore,  that  unless  the  given 
arc  exceed  60°,  it  is  itself  the  path  required. 

259,  Let  us  next  consider  the  case  in  which  the  given  arc 
AC  exceeds  60°  ;  that  is,  in  which  2sin(^  is  less  than  unity. 

Here  the  first  of  eqs.  (11)  is  satisfied  when  sin/'=  2  sin  a; 
and  as  v  then  becomes  a  maximum,  it  may  evidently  have  the 
same  value  twice.  But  now  the  value  oiy  —  b  just  given  would 
become  positive  before  we  reach  the  point  B,  and  so  a  part  of 
(^^  would  become  negative  if  we  suppose  the  path  to  termi- 
nate with  a  portion  of  the  circular  arc  through  B,  which  is 
inadmissible. 

We  conclude  then,  in  this  case,,  that  the  required  path  must 
consist  of  the  circular  arc  AR  and  the  cycloidal  arc  RB  tan- 
gent to  the  circle  internally  at  R.  Then  y  —  b  will  be  nega- 
tive for  the  whole  circular  arc  AR.  For  as  the  cycloid  is 
tangent  to  the  circle  internally  at  R,  its  radius  of  curvature 
must  at  that  point  be  less  than  r;  that  is,  since  the  radius 
is  twice  the  normal,  2RT<r,  so  that  0T>  RT,  Whence 
OT  sin  TOE  =  EF  =  b  is  greater  than  RT sin  RTF  or 
RF  sin  TOE ;  that  is,  ^  >  y  and  y  —  b  is  negative. 

260,  We  must  now  show  that  a  cycloid  can  be  drawn 
tangent  to  the  circle  internally  at  R  and  passing  through  B. 


DISCONTINUOUS  SOLUTIONS.  32/ 

First,  assume  D  —  CB  z=  r  —  b,  putting  the  vertex  at  B. 
Then,  since  2(r  —  b),  the  radius  of  curvature  at  B,  is  greater 
than  r,  the  cycloid  will  be  tangent  to  the  circle  externally 
at  B.  But  by  taking  D  sufficiently  large,  the  cycloid  still 
passing  through  B,  we  can  cause  the  cycloid  to  fall  entirely 
within  the  circle,  and  then  by  diminishing  D,  while  retaining 
^  as  a  cycloidal  point,  we  must  arrive  at  a  value  of  D  which 
will  cause  the  cycloid  to  become  tangent  to  the  circle  before 
cutting  it,  and  this  point  of  contact  will  be  neither  at  A  nor  B. 
For  at  A,  y'  for  the  cycloid  is  infinite,  while  for  the  circle  it  is 
not ;  and  at  B^  y'  for  the  circle  is  zero,  while  for  the  cycloid  it 
is  not. 

Now  as  the  solution  is  real,  it  is  unnecessary  to  discuss  the 
value  of  D  or  the  position  of  the  point  of  contact  R,  or  of  the 
cusps  on  A  C. 

261.  No  natural  discontinuity  presents  itself  in  the  discus- 
sion of  Prob.  II.,  since,  if  the  two  fixed  points  be  not  in  the 
vertical  nor  in  the  horizontal  line,  we  can  cause  a  cycloid  to 
pass  through  them  both,  and  have  its  cusps  on  the  horizontal 
line  through  the  upper  point.  Neither  can  there  be  admissi- 
ble but  unnecessary  discontinuity  of  the  kind  discussed  in 
Prob.  XXXIX.  For  if  there  could  be  two  cycloidal  arcs 
meeting  at  any  point,  they  must,  as  we  have  seen,  both  have 
their  cusps  on  the  horizontal  through  the  point  from  which 
the  particle  starts,  and  must  also,  as  appears  from  Art.  256, 
be  tangent.  Moreover,  from  Art.  25,  the  fundamental  equa- 
tion is  j(i  +y^)  =  ^  =  i^;  and  since  y'  has  but  one  value  at 
the  point  of  contact,  D  can  have  but  one  value  there  for  the 
two  cycloids,  and  the  cycloidal  arcs  must  therefore  be  gen- 
erated by  the  same  circle  rolling  on  the  same  horizontal. 
Now  as  y'  in  any  cycloid  can  have  a  given  value  but  once, 
these  arcs  have  also  their  cusps  in  common ;  that  is,  there  are 
not  two  cycloidal  arcs  at  all. 


328  CALCULUS  OF    VARIATLONS. 

Section  X. 

OTHER  METHODS   OF    VARIATLONS. 

262.  Hitherto,  whether  employing  plane  or  polar  co- 
ordinates, we  have  ascribed  variations  to  the  dependent  vari- 
able only  and  its  differential  coefficients,  adding  also,  when  a 
change  in  the  independent  variable  is  necessary,  an  increment 
to  its  limiting  values  only.  This  method,  which  has  been 
adopted  by  the  two  most  elaborate  English  writers.  Profs. 
Jellett  and  Todhunter,  as  also  by  the  chief  German  writer, 
Strauch,  is  undoubtedly  the  best.  But  many  writers  vary 
the  independent  variable  also  throughout  the  whole  definite 
integral ;  and  as  the  reader  will  be  likely  to  meet  with  this 
method,  the  present  work  would  be  incomplete  if  it  did  not 
explain  this  method  sufficiently  to  enable  him  to  follow  the 
solution  of  any  problem  in  which  it  may  be  employed. 

First  Method. 

263.  Suppose  we  assume  the  equation 

U^  r^Vdx,  ■     (i) 

where  Fis  any  function  of  ;r,  j,  /,  etc.,  and  suppose  j;/  to  be- 
come the  ordinate  of  some  primitive  curve.  Then,  by  varying 
^in  the  most  general  manner,  we  can  pass  to  any  curve  which 
can  be  derived  from  the  first  by  infinitesimal  changes  in  .t'o,  x^y 
y,  y,  etc. 

But  we  may  also  pass  to  the  same  derived  curve  by  mov- 
ing, without  change  of  value,  any  ordinate  of  the  primitive 
curve  an  infinitesimal  distance  djtr  along  the  axis  of  .v,  and  then 
varying  it  so  as  to  make  it  become  the  ordinate  of  the  derived 
curve  for  the  new  abscissa  x  +  ^-^.  In  this  method  (^j/,  ^y, 
etc.,  will  mean  the  difference  between/,/',  etc.,  for  the  primi- 


OTHER  METHODS   OF    VARIATIONS.  329 

tive  curve,  and  corresponding  to  the  abscissa  x,  and  the  same 
quantities  for  the  derived  curve  corresponding-  to  the  abscissa 
X  -\-  ^x.  Of  course  for  any  given  value  of  x  we  may  suppose 
f^x\.o  have  either  sign,  or  to  vanish  ;  and  it  is  evident  that  when 
the  hmits  are  to  be  fixed,  the  latter  supposition  must  be  made 
regarding  the  quantities  ^x^  and  Sx^, 

264,  We  are  led,  then,  to  inquire  what  will  be  the  expres- 
sion for  dU,  when  x  also  is  regarded  as  capable  of  variation 
throughout  the  definite  integral  U. 

In  (i)   change  x  into  x-{-dx,  y  into  y -\-  Sy,  etc.,  and  let 

U'  =  [/+  6U         and     V  =  V+  dF  (2) 

be  the  new  values  of  [/  and  V.  Then,  observing  that  dx  will 
become,  by  being  varied, 

Sd.v  =  -^_(.v  +  ^-r)dx,  (3) 

we  shall  have 

^'  =  X''^"^^"+<'-)^---  (4) 

Whence 

^^^         dx  ^^^ 

=jiy  ^^+ '  ^)  i  (- + <^-)  '^'^  -£'  ^'^--  (5) 

This  is  exact ;  but  approximating  to  the  first  order  only,  we 
have 


330  CALCULUS  OF    VARIATIONS. 

where  brackets  denote  the  complete  differential  coefficient  of 
F;  that  is, 

Moreover,  it  is  evident  that,  to  the  first  order, 

6V^  Mdx  +  Ndy-^Pdy'  +  etc.  (9) 

Hence  (6)  becomes 

du^  K6x,  -  VM.+X^' )  my+p^y+  Qdy^+  etc. 

-  {JVy  +  Py"  +  Qy'"  +  etc.)  dx  \  dx,  (10) 

266.  But  the  formulae  hitherto  emplo3^ed  for  Sy\  dy'\  etc., 
will  not  now  hold  true,  so  that  we  must,  before  we  can  fur- 
ther transform  (10),  ascertain  what  will  be  the  values  of  these 
quantities  under  the  present  supposition. 

First,  in  y  change  x  into  x-\-dx  and  y  into  y  -\-  dy^  and 
we  have 

sy-^Ay±M-yJ       ^/  -y     . 

"^  dx 
dSy      y'ddx 
.    .    dx  dx  ,  _  fdSy      y'dSx\  f      ,    dSx\  —  ^      ,     v 


^  ~^  dSx        ^~\dx  dx  IV    '     dxj 

dx 


which  is  exact ;  and  to  approximate  to  any  order  required  we 
have  only  to  develop  sufficiently  the  second  factor.  Thus,  to 
the  second  order, 


OTHER  METHODS  OF    VARIATIONS.  331 

or,  omitting  the  terms  of  the  second  order. 

To  obtain  to  the  first  order  the  value  of  ^y"  we  have  only  to 
substitute  in  (13)/  ior  y,y"  ior  y',  and  y  ior  /\  so  that 

6y"^^{Sy-y"Sx)+y"'6x  =  £,  {^Sy  - /Sx)+ /"Sx.  (14) 

The  Greek  letter  &?  (omega,  or  o)  is  usually  put  for  6y  —  y'6x. 
Then  we  shall  find 


.y  =  g +/'..,      sy'  =  '-^+y",., 


(15) 


which  equations  are,  of  course,  true  to  the  first  order  only. 

266.   Now  substituting  in  (lo)  the  values  of  ^/y^y^  etc., 
derived  from  (15),  that  equation  will  become 

dU=  V,dx-  V.dxo  +Xy^'^  +  ^^'  +  Q^"  +  etc.)^;tr,      (16) 

where  od'  =  — -,  etc.     Here  oo,  od' ,  od",  etc.,  take  the  place  of 

ax  ^ 

Sy,  dy',  6y'\  etc.,  in  the  former  method,  so  that  integrating  by 
parts,  as  in  that  method,  we  shall  obtain 

dU  =  V,Sx^  —  V,Sx,  +  k^oo^  —  h,0D^  +  i,Go^'  —  i,oo^  +  etc. 

+£?^^-  P'  +  Q"  -  etc>^^,  (17) 

where  the  coefficients  of,  (y^,  od^,  go/,  etc.,  are  the  same  as  those 
of  ^y„  Sy,,  Sy/,  etc.,  in  equation  (5),  Art.  36,  /i,  t,  etc.,  being 


332  CALCULUS  OF  VARIATIONS. 

used  as  in  equation  (7),  Art.  37  ;  while  the  coefficients  of  Godx 
and  Sydx  are  also  identical. 

Moreover,  since  dx^,  dx^  and  ^x^,  dx^  mean  the  same  thing 
in  the  two  methods,  it  appears  that  (^'^in  this  case  is  the  same 
in  form  as  the  most  general  variation  of  U  found  by  the  other 
method,  go  taking  the  place  of  fy. 

267.  Suppose,  now,  we  wish  to  discover  by  this  method 
the  conditions  which  will  maximize  or  minimize  U.  Then  it 
will  appear,  by  the  same  reasoning  as  before,  that  (^C/to  the 
first  order  must  vanish,  while  the  terms  of  the  second  order 
must  preserve  an  invariable  sign,  becoming  negative  for  a 
maximum  and  positive  for  a  minimum.  Hence  (17)  may  be 
written 

SU=L,-L,+X^'Mcodx 

=  L,-L,  ^£^'Mdydx  —fJ'MySxdx  =  o.      (18) 

Therefore  the  coefficients  of  Sydx  and  dxdx  are  so  related 
that  if  one  vanish  the  other  must  vanish  also,  unless,  indeed, 
y  should  become  zero  throughout  the  curve. 

Now  ^x  and  fy  under  the  integral  sign  are  entirely  inde- 
pendent of  any  conditions  which  those  quantities  may  be  re- 
quired to  fulfil  at  the  limits,  and  therefore  we  must  have 

L^  —  L^  =  o       and     /     Moodx  =  o.  (19) 

But  QD,  like  Sy,  is  wholly  in  our  power,  Avhile  M  \s  not,  so  that 
we  must  necessarily,  as  before,  suppose  M  to  vanish,  and  we 
can  obtain  no  additional  equation  by  considering  separately 
the  integrals  in  the  last  member  of  (18). 

268.  Let  us  now  briefly  consider  the  terms  at  the  limits. 
Suppose,  in  the  first  place,  x^^  x^j  y^y  /o>  •  •  •  •  Jo^**""'^  to  be 


OTHER  METHODS  OF   VARIATIONS.  333 

fixed ;  that  is,  to  have  no  variation.  Then  qd^,  gj^,  go/,  etc., 
and  Sjt:^  and  ^x^,  will  severally  vanish.  For  let  y"*)  be  an}'  dif- 
ferential coefficient  of  j  not  higher  than  y**  - 1).  Then  we  have 
at  either  limit 

and  dx  being  zero  at  either  limit,  we  have  for  that  limit 

d^QD  ,. 

—   Gj(^)  =  O. 


dx 

Hence,  in  this  case,  L^  —  L^  will  vanish,  and  we  must  deter- 
mine the  271  constants  as  we  did  formerly  when  all  the  limit- 
ing values  were  fixed. 

Let  us  next  suppose  x^  and  x^  only  to  be  fixed.  Then,  at 
either  limit,  od  =  (^J/,  od'=  ^y,  oo"^  dy" ,  etc.,  and  assuming  these 
quantities  to  be  unrestricted,  h^,  h^,  i^,  /„,  etc.,  must  severally 
vanish,  which  are  the  same  conditions  for  the  determination 
of  the  constants  as  we  would  have  under  the  same  supposi- 
tion by  employing  the  other  method.  Neither  can  we  ob- 
tain any  additional  equations  by  putting  for  cj,  gd\  etc.,  their 
values,  and  then  making  Sx  at  the  limits  vanish.  If  we  make 
the  limiting  values  of  y  also  invariable,  c^^  and  gd^  will  vanish, 
all  the  other  conditions  remaining  as  before,  so  that  we  shall 
only  lose  the  equations  //,  ==  o  and  h^  =  o,  which  will  be  re- 
placed by  the  conditions  that  y^  and  y^  must  have  given  values. 

Proceeding  similarly,  it  will  appear  that  when  x^  and  x^ 
are  fixed,  the  same  equations  for  the  determination  of  the  271 
arbitrary  constants  arising  from  the  integration  of  the  equa- 
tion M  =  o  will  be  obtained  as  would,  under  the  same  sup- 
position, have  been  found  by  the  other  method. 

Let  us,  in  the  last  place,  suppose  that  x^  and  x^  are  also 
variable.  Then,  if  no  restriction  be  imposed  upon  any  of  the 
variations,  we  shall  have,  besides  the  equations  already  ob- 
tained, V^  =  o  and  V^  =  o,  and  we  shall  find  that  we  cannot 


334  CALCULUS  OF    VARLATLONS. 

obtain  any  other  equations.  Here  the  conditions  are  the 
same  as  those  noticed  in  Art.  JJ,  and  the  211  -f  2  equations 
cannot  in  general  be  satisfied. 

But  suppose  that,  as  in  Prob.  IX.,  the  extremities  of  the 
required  curve  are  to  be  confined  to  two  fixed  curves  whose 
equations  are,  as  in  Art.  69,  jj/=/and  y  —  F,  f  and  i^  being 
functions  of  x.  Here  Sy  has  not  the  same  meaning  as  in  the 
former  method,  so  that  equations  (10),  Art.  69,  or  rather  equa- 
tions (2),  Art.  yG,  will  not  now  be  applicable.  But  it  is  evi- 
dent that  now  Sy^  ^z^f^dx^  and  dy^  =  FJSx^;  so  that  we  shall 
now  have  at  the  upper  limit 

^r  =  (//  -  j/)  ^^^,  ^/  =  ]  #-  ir  -  /)  \     ^^.         (20) 


dx^-"        -^  M 


and  similar  equations  in  F  hold  for  the  lower  limit.  Now 
observing  that  dx^  and  dx^  here  mean  the  same  thing  as  dx^ 
and  dx^  in  the  other  method  when  used  to  change  the  limiting 
values  of  x,  we  see  from  equations  (2),  Art.  76,  that  for  either 
limit  we  must  substitute  the  same  thing  for  Sy  in  the  first 
method  as  for  gd  in  the  second,  and  the  same  thing  for  dy'  in 
the  first  as  for  gd'  in  the  second ;  so  that  the  coefficients  in- 
volved must  evidently  be  the  same  in  both  methods.  Hence 
we  must  always  obtain  by  either  method  precisely  the  same 
equations  of  condition  at  the  limits. 

269.  Thus  it  will  be  seen  that  the  results  obtained  by  the 
two  methods  are  the  same,  whether  as  regards  the  general 
solution,  or  the  conditions  which  must  hold  at  the  limits,  and 
that  nothing  is  gained  by  the  latter  method,  while  the  labor 
of  obtaining  the  required  results  is  somewhat  increased. 
This  disadvantage  will  become  still  more  obvious  when  we 
seek  to  examine  the  sign  of  the  terms  of  the  second  order. 
We  shall  not,  however,  enter  upon  this  examination  further 
than  to  observe  that  we  must,  in  finding  these  terms,  be  care- 
ful not  to  reject  any  of  the  terms  of  the  second  order.     Thus, 


OTHER  METHODS  OF   VARIATIONS,  335 

after  having  approximated  to  the  second  order  in  equation  (5), 
if  we  employ  (13)  and  (14)  in  transforming  the  terms  of  the 
first  order,  we  must  remember  that  the  value  of  Sy  which 
we  now  require  is  given  by  (12),  and  that  (13)  and  (14)  are 
not  sufficiently  accurate,  and  that  we  must  therefore  add  to 
the  terms  already  assigned  to  the  second  order  those  which 
are  neglected  in  the  first  by  the  use  of  (13)  and  (14) ;  and  it  is 
easy  to  see  that  this  will  generally  involve  us  in  much  diffi- 
culty. 

It  is  believed  that  the  foregoing  account  of  the  present 
method  will  be  found  sufficient  to  enable  the  reader  to  follow 
any  solution  which  may  be  presented,  which  is  all  that  is 
necessary,  since  its  adoption,  as  a  mode  of  original  investiga- 
tion, cannot  be  advised. 

Second  MetJiod. 

270,  The  method  which  we  next  proceed  to  explain  pos- 
sesses oftentimes  decided  advantages,  particularly  when  we 
come  to  consider  problems  involving  three  co-ordinate  axes, 
and  is  moreover  that  which  is  adopted  by  Prof.  Jellett  in  the 
discussion  of  geometrical  problems.  As  we  shall  be  obliged 
to  consider  it  at  some  length,  the  reader  will,  we  think,  most 
easily  comprehend  its  nature  and  use  by  the  consideration  of 
an  example. 

Problem  XLV. 

//  is  required  to  discover  the  co?iditions  which  will  maximize  or 
minimize  the  expressioji   U  ^^  j     vVi  -\- y''^dx,  zvhere  v  is  any 

function  of  x  and y  only,  and  constants ,  the  limits  being  fixed  or 
variable. 

Now  assuming  s  as  the  arc  of  the  required  primitive  curve, 
C/'may  be  written 

u^  r^vds.  (i) 


33^  CALCULUS   OF  VARIATIONS. 

Let  ab  be  the  required  arc,  and  on  it  take  at  pleasure  any  points 
c,  d,  e,  etc.,  and  regard  these  points  as  knots  or  spots  upon  a 
flexible  cord.  Then,  when  we  make  any  infinitesimal  altera- 
tion in  the  form  of  ab,  the  arcs  ac,  ad,  ae,  etc.,  will  undergo  no 
change  in  length,  but  the  co-ordinates  of  the  points  c,  d,  e,  etc., 
will  in  general  undergo  an  infinitesimal  change. 

But  the  arcs  ac,  ad,  etc.,  are  any  values  of  s,  measured  from 
a,  so  that  it  appears  that  we  can  pass  from  ab  to  any  derived 
curve  by  varying  x  and  j/  in  (i),  while  regarding  s,  and  there- 
fore ds,  as  undergoing  no  variation. 

27!.  Taking  the  variation  of  (i)  under  this  supposition, 
we  have 

But  (2)  gives  the  variation  of  U  only  under  the  supposition 
that  we  need  not  make  any  change  in  the  length  of  the  primi- 
tive curve,  which  is  not  usually  the  case.  For  suppose  the 
required  curve  be  conditioned  to  always  connect  two  fixed 
points  or  two  fixed  curves.  Then  if  we  vary  ab  without  pro- 
ducing any  change  in  its  length,  we  shall  in  reality  reduce  the 
problem  to  one  of  relative  maxima  and  minima,  in  which  the 
length  of  s  is  to  be  fixed,  and  in  which,  as  we  have  already 
shown,  the  form  of  the  derived  curve  cannot  be  wholly  unre- 
stricted. If,  then,  the  problem  be,  as  we  have  assumed,  one  of 
absolute  maxima  and  minima — that  is,  if  we  are  required  to 
vary  the  form  of  ab  in  the  most  general  manner  consistent 
with  the  method  of  variations — the  arc  of  the  derived  curve 
connecting  the  given  points  or  given  curves  will  not  neces- 
sarily have  the  same  length  as  ab.  Still  it  is  not  necessary  to 
vary  s  or  ds  under  the  integral  sign,  because  we  can  evidently 
pass  from  ab  to  any  derived  curve  AB  by  first,  before  varying 
ab,  giving  to  it  increments  or  decrements  at  a  and  b  so  as  to 
obtain  a  new  arc  equal  in  length  to  AB,  and  then  varying  the 
form  of  this  new  arc  in  the  most  general  manner. 


OTHER  METHODS  OE   VARIATIONS.    .  337 

But  as  these  increments  must  be  infinitesimal,  we  may  de- 
note them  by  ds^  and  ds^.  Now  if  in  (i)  we  change  the  hmits 
into  s^  -["  ^^^  ^^<^  -^1  +  ^^v  we  may  find  approximately  the  change 
which  will  result  to  U  in  precisely  the  same  manner  as  if  the 

expression  were  U  =^  I  Vdx,  and  x^  and  x^  only  were  to  be 
varied.     Hence  this  change  will  be 

V,  ds,  -  V,  ds,  +  \  \^^f'^  -  \  \j^f'^'  +  ^^^^  (3) 

where  brackets  denote  the  total  differential  coefficients.  But 
we  wish  to  find  t^^to  the  first  order  only,  so  that  we  may 
write,  as  the  new  value  of  U, 

U'  ^^  U  -\-  z\ds^  —  v^ds^  =  i\ds^  —  v^ds^  -|-    /     vds.         (4) 

If  now  we  vary  the  form  of  the  arc  in  the  most  general 
manner,  and  suppose  U'  to  become  U'^,  U"  will  exceed  U'  by 
the  second  member  of  (2)  increased  by  dv^ds^—  Sv^ds^.  Hence, 
observing  that  the  last  two  terms,  being  of  the  second  order, 
must  be  rejected,  wc  shall  find 

u"-u=su=  V, ds, - ., ds,  +X"  1 J s-- +j/y\dS'    (5) 

which  is  the  form  of  (^"^  which  we  must  in  general  employ 
whether  the  curve  be  required  to  connect  two  fixed  points  or 
two  fixed  curves. 

272.  As  Sx  and  5y  now  denote  the  changes  which  the  co- 
ordinates of  any  point  when  regarded  as  fixed  on  the  arc,  like 
a  knot  on  a  cord,  would  undergo,  owing  to  any  infinitesimal 
alteration  in  the  form  of  the  arc,  it  will,  we  think,  appear  after 
a  little  reflection  that  we  cannot  regard  ^x  and  Sy  as  entirely 
independent,  although  we  cannot  state  explicitly  the  nature 
of  the  relation  subsisting  between  them.     We  can,  however, 


338  CALCULUS  OF  VARIATIONS. 

easily  assure  ourselves  that  they  are  not  independent.     For  if 

they  were,  then,  since  d  6^  must  vanish  in  order  that  f/may  be 

y-i-idv  jydv 

a  maximum  or  a  mmimum,  we  would  have  --  =  o  and  --  =  o. 

ax  ay 

Whence 

dv       dvdy  _  Vdv~\  _ 
dx      dy  dx       VjdxA 

Therefore  we  would  have  as  a  condition  necessary  to  a  maxi- 
mum or  a  minimum  -^^  =  a  constant,  which  is  false,  since  in 
Prob.  VII.  we  have 

u  ^  ly  ^Y^y- dx  ^  ly  ds, 

and  y  is  not  constant. 

If  we  could  express  ^y  explicitly  in  terms  of  Sx  and  other 
quantities,  we  might  eliminate  one  of  the  variations,  and  then 
equate  the  coefficient  of  the  remaining  variation  to  zero. 
But  as  this  cannot  be  done  in  the  present  case,  an  ingenious 
method  of  overcoming  this  difficulty  has  been  devised  by  La- 
grange, which  we  now  proceed  to  apply,  reserving  a  general 
explanation  of  this  method  until  the  reader  has  become  some- 
what familiar  with  its  spirit. 

273.  We  have  always,  whether  along  the  primitive  curve 
ab  or  the  derived  curve  AB,  ds"  —  dx^  -\-dy^^  so  that 

^-+/^_i=o,  (6) 

where  accents  will  denote  differentiations  with  respect  to  s ; 
and  as  this  equation  must  always  hold,  it  follows  that  the  vari- 
ation of  its  first  member — that  is,  the  change  which  that  mem- 
ber will  experience  when  we  change  x  into  x -\-  ^x  and  y  into 
y-{-  ^y,  s  remaining  unaltered — will  be  zero.  Hence  we  must 
have 

x'dx'  -\-y'dy'  =  o.  (7) 


OTHER  METHODS  OF   VARIATIONS.  339 

Then,  when  we  change  x  into  x  -\-  Sx,  the  new  value  of  x'  is 
-(.+  <J.)    or    .+  —  . 


Hence 

ds 


^-'  =  ^-  •  (8) 


Similarly,  when  we  change  x'  into  x'  -f-  Sx\  the  new  value  of 

x"  is 

--(.+<y.)     or    .   +--. 


Whence 

ds  ds 


<^^  =-7r  =  ^7^-  (9) 


In  the  same  way  we  shall  find 

<jy  =  ^,  Sy     =-^,        etc.  (10) 

Now  these  formulae  are  analogous  to  those  in  Art.  9,  and 
are,  unlike  those  obtained  in  the  preceding  method,  exact. 
Moreover,  it  is  evident  that  equations  similar  to  those  just 
obtained  must  hold  when  we  have  any  number  of  variables 
X,  y,  z,  u,  etc.,  all  dependent  upon  the  same  independent  vari- 
able, which  is  itself  incapable  of  receiving  any  change  from 
being  varied,  the  limiting  values  only  being  supposed  to  -be 
susceptible  of  an  increment. 

274.  Now  because  (7),  (8)  and  (10)  are  true,  we  may  write 


t/So        1 


dSx    ,    ,  ,  dSv  \   J  ,     , 

—  ^ly-^.\ds^O,  („) 


340  CALCULUS  OF  VARIATIONS. 

where  /  may  be  any  finite  quantity  we  please,  either  constant 
or  variable.  Transforming  (ii)  in  the  usual  way,  and  adding 
the  resulting  equation  to  the  value  of  dUm  (5),  we  have 

dU=  v,ds,  —  v,ds,  +  l^x'dx  +  y'6y\  —  IJ^x'Sx  -^y'^y), 

=  L,-L,^£l\MSx^NSy\ds,  (12) 

where  the  suffixes  x  and  y  denote  partial  differentiation  with 
respect  to  those  quantities. 

As  we  have  now  introduced  into  SU  the  only  connecting 
equation  between  Sx  and  8y,  and  have  reduced  the  result  as 
far  as  possible,  it  will  appear,  by  reasoning  precisely  like  that 
hitherto  employed,  that  since  an  unrestricted  integral  cannot 
equal  a  given  function  of  quantities  relating  to  the  limits  only, 
we  must  have 

Z,-Z,  =  o,        £l'\MSx^NSy\ds  =  o.  (13) 

Now  since  /  is  in  our  power,  suppose  it  to  be  so  taken  as  to 
cause  M  to  vanish  throughout  U.     Then  the  second  of  equa- 

tions  (13)  will  become  /     NSy  ds  =  o;  and  as  Sy  is  evidently 

entirely  independent  of  N,  this  equation  can  only  be  satisfied 
by  making  N  also  vanish  throughout  U, 

275.   We  have  then  the  equations 

Vx  —  ilx')'  =:  O  =  Vx  —  Ix"  —  I'x'^  ) 

\  (14) 

Vy-{ly')'  =  o  =  v^-l/-l'y'.) 


OTHER  METHODS  OF   VARIATIONS.  341 

Multiply  the  last  members  by  x'  and  y'  respectively,  and  add. 
Then,  observing  that 

v,x'^Vyy'=^v',  x'x"-\-y'y"^0,  .r"+y' =  i,  (15) 
we  shall  obtain 

v'  —  I'  ^o        and     I  —  v  -\-  c.  (i6) 

276.  Before  proceeding  further  we  must  fully  determine 
/  by  ascertaining  the  value  of  c,  which  can  be  done  by  means 
of  the  terms  at  the  limits,  which  we  will  next  examine. 

To  prepare  our  way,  we  observe,  first,  that  it  is  immaterial, 
in  passing  from  a  primitive  to  a  derived  curve,  whether  we 
first  increase  the  limits  by  the  positive  or  negative  increments 
^^0  and  ds^,  and  then  vary  the  new  arc,  or  first  vary  the  form 
of  the  entire  arc,  and  then  add  these  same  increments  to  its 
extremities. 

In  the  second  place,  the  increments  which  would  result  to 
X  or  y\n  passing  along  any  infinitesimal  arc,  ds^  or  ds^,  while  it 
belongs  to  the  primitive  curve,  and  also  after  it  has  undergone 
an  infinitesimal  change  of  form,  but  none  in  length,  and  has 
assumed  its  new  position  as  a  part  of  the  derived  curve,  can- 
not differ  by  an}^  term  of  the  first  order,  although  they  may 
differ  by  a  term  of  the  second  order. 

277.  Suppose,  first,  that  the  curve  is  to  connect  two  fixed 
points  A  and  B,  the  required  primitive  curve  being  ab,  so 
that  before  it  is  varied  a  will  be  at  yi,  and  /^  at  B,  and  consider 
the  upper  limit.  At  b  add  a  positive  or  negative  increment 
ds^,  and  denote  the  new  arc  by  ac.  Also  let  x^  -f  <^-^i  ^^d 
y^  +  dy,  be  the  co-ordinates  of  c ;  then  it  is  evident  that  the 
difference  between  the  co-ordinates  of  b  and  c — that  is,  the  in- 
crements which  would  result  to  x  and  y  by  passing  along  the 
arc  from  c  to  b — must  be  —  dx^  and  —  dy^  or  —  x^  ds^  and 
—  yl  ds^.  Now  vary  the  form  of  ab.  Then  the  pomt  ^  will 
assume  a  new  position  whose  co-ordinates  will  be  x^  -f~  ^^\  ^"^^ 


342  CALCULUS   OF    VARIATLONS. 

Ji  +  ^y-ii  while  the  point  c  will  now  fall  upon  B.  Hence  Sx^ 
and  Sy^  are  the  increments  which  x  and  y  receive  as  we  pass 
from  ^  to  ^  on  the  derived  curve.  Therefore,  as  the  arc  Bb 
on  the  derived  curve  was  the  arc  be  or  Be  on  the  primitive 
curve,  having,  without  change  of  length,  merely  altered  its 
position  and  form  infinitesimally,  it  appears,  by  the  second 
remark  of  the  last  article,  that,  to  the  first  order,  we  must 
have 

Sx^  =  —  x^'ds^y         dy^  =  —  y^ds^ ;  (17) 

and  similar  equations  would,  of  course,  hold  at  the  lower  limit. 

278.  Next,  suppose  the  required  curve  is  to  connect  two 
fixed  curves  whose  differential  equations  are  dy^f'dx  and 
dy  =  F'dx,  and  consider  the  upper  limit. 

Let  the  required  primitive  curve  cut  the  fixed  curves  at  b 
before,  and  at  B  after  it  has  been  varied.  Then  we  can  easily 
find  the  co-ordinates  of  B  from  the  first  remark  of  Art.  276. 
For  when  we  vary  the  primitive  curve,  the  co-ordinates  of  the 
extremity  in  question  will  become  x^  -\-  Sx^  and  y^  +  Sy^ ;  and 
if  now  to  this  extremity  we  add  the  positive  or  negative 
increment  ds^,  denoting  by  dx^  and  dy^  the  corresponding 
increments  of  x  and  y,  we  shall  reach  B,  whose  co-ordinates 
must  therefore  be  x^  -j-  ^-^1  +  '^■^i  ^^^  j,  -\-  Sy^  -\-  dy^.  Sub- 
tracting x^  and  7i,  we  find  the  changes  which  x  and  y  experi- 
ence as  we  pass  from  b  to  B  along  the  fixed  curve  to  be 
respectively    Sx\  +  dx^    or    dx^  -f"  x^'ds^    and    dy^  -\-  dy^    or 

But  the  increment  which  results  to  y  in  passing  along  the 
arc  bB  must  be  /^  times  that  which  results  to  x\  so  that  we 
must  have 

Sy^+y:ds^f,\dx,  +  x:dsy,     '  (18) 

and  a  similar  equation  in  F^  can  evidently  be  obtained  for  the 
lower  limit.  Of  course  these  equations,  like  (17),  are  true  to 
the  first  order  only,  because  we  have  estimated  ds^  along  the 


OTHER  METHODS   OF   VARIATIONS.  343 

derived  curve,  whereas  it  should  be  taken  along  the  primi- 
tive curve. 

279.  Let  us  now  consider  the  terms  at  the  limits  in  (12), 
first  supposing  the  required  curve  is  to  connect  two  fixed 
points.  Here  substituting  for  dx^  and  dy^^  and  also  for  Sx\ 
and  c5jj/„,  their  values  from  (17),  and  observing  equation  (6), 
we  shall  obtain 

{v  —  l\  ds,  —  {v  —  l\  ds,  =  o.  (19) 

But  ds^  and  ds^  are  entirely  independent,  so  that  their  coeffi- 
cients must  severally  vanish.  Hence  we  have  /,  =  v^,  and  c 
in  (16)  must  become  zero,  giving  us  /=  7^  throughout  the  in- 
tegral. 

Next  suppose  the  curve  is  to  connect  two  fixed  curves  as 
in  the  last  article,  and  consider  the  upper  limit.  Substituting 
the  value  of  Sy^  found  from  (18),  this  limit  gives 

V,  ds,  +  I,  x;Sx,  +  l,y,\f'Sx  +  fx'ds  -  yds\  ;  (20) 

and  a  similar  equation  will  hold  at  the  lower  limit.  Now 
these  two  limiting  equations  must  be  absolutely  independent, 
because  we  may  suppose  one  extremity  of  the  required  curve 
to  be  absolutely  fixed.  We  must,  therefore,  equate  (20)  to  zero. 
Now  it  will  appear  upon  a  little  reflection  that  dx^  and  ds^ 
must  be  also  entirely  independent,  so  that  we  may  equate 
their  coefficients  severally  to  zero.  Hence,  if  l^  be  not  zero, 
(20)  will  give 

^.  +  hyM'^'  -y\  =  o,      ^/ +/>/  -  o.        (21) 

Substituting  in  the  first  of  these  equations  for  //jk/  its  value, 
—  x/,  found  from  the  second,  and  observing  equation  (6),  we 
obtain,  as  before,  ^j—  /^  =  o ;  so  that  here  also,  as  appears  from 
(16),  v  =  L 

If  /j  should  become  zero,  then,  since  neither  x/  nor  y/  can 
become  infinite,  the  upper  hmiting  terms  would  reduce  to 


344  CALCULUS  OF    VARIATLONS. 

v^ds^  =  o,  SO  that  v^  must  also  vanish.  Hence,  here  also,  c  in 
(i6)  vanishes,  and  we  therefore  have  always  v  —  L 

280.  The  reader  of  Prof.  Jellett's  work  will  observe  that 
in  Chapter  IV.,  in  which  he  adopts  this  method,  he  has,  in 
giving  the  terms  at  the  limits,  uniformly  omitted  the  terms 
V^ds^  —  V^ds^y  and  this  omission  has  led  him  into  an  unsatis- 
factory method  of  determining  the  constant  c,  which  is  in  his 
book  a,  and  which,  as  we  have  seen,  can  be  determined  regu- 
larly by  the  equations  at  the  limits.  (See  Todhunter's  His- 
tory of  Variations,  Art.  348.)  It  happens,  however,  that  his 
results  in  discussing  by  this  method  the  conditions  which 
must  hold  at  the  limiits  are  in  every  case  correct,  although 
the  method  by  which  they  are  obtained  is  certainly  not 
strictly  so.  The  reader  will  find  it  profitable  to  verify  this 
latter  assertion,  which  is  made  upon  the  authority  of  the 
author  alone. 

281.  Let  us  now  return  to  the  general  solution.  Putting 
V  for  /in  the  last  members  of  (14),  we  have 

vx'',         Vy  —  yv'  =  vy".  (22) 

Now  in  these  equations  multiply  v^  and  Vy  by  ^''^+y^  which 
is  unity,  and  put  in  each  for  v'  its  value  from  the  first  of  equa- 
tions (15).     Then,  reducing  and  factoring,  we  shall  obtain 

y'i:i'xy'  —  'Vyx')  =  vx",         x'(vyx'  —  v^y')  =  v/\         (23) 

Multiplying  the  first  of  these  equations  by  y\  the  second  by 
x\  and  subtracting  the  second  from  the  first,  remembering 
equation  (6),  we  have 

'^x/—  'Vyx'  =.  V  {y'x"—  x'y").  (24) 

Let  r  be  the  radius  of  curvature.  Then  we  know  that 
y'x"  —  x'y"  ~  -.     Hence  we  may  Avrite 


OTHER  METHODS  OF   VARIATIONS.  345 

-  —  -  {vxy'  —  Vyx')  — {v^  COS  A  4-  Vy  cos  B\        (25) 

r       V  ^ 

where  A  is  the  angle  which  the  normal  makes  with  the  axis  of 
x^  and  B  the  acute  angle  Avhich  it  makes  with  the  axis  of  y. 

It  is  impossible  to  proceed  further  with  the  solution  so 
long  as  the  form  of  v  is  wholly  undetermined ;  but  equation 
(25)  will  enable  us  to  solve  many  problems  with  great  ease,  as 
we  will  now  show. 

282.  Consider  Prob.  I.     Here  U=J^  ^ ds,  so  that  v  =  i, 

Vx  =  o,  %  =  o.     Therefore  equation  (25)  gives  —  =  o.   Hence, 

r  being  infinite,  the  solution  must  be  a  right  line. 

Turn  next  to  Case  2,  Prob.  II.     Here  ^^may  be  Avritten 

XSi    (Is                                                I                                                     I 
— —y   SO   that   V  =:  -— =-,  Vx  =  o,  Vy  = 3,  and  (25) 
-     \y                              Vy                               V" 

prives  —  = and   r  =  21/  sec  B.      Let   ;/   be   the    normal. 

Then  7t  ^=  y  sec  B  and  r  =  2;/,    which    is   known   to    indicate 
that  the  required  curve  must  be  a  cycloid. 

In  the  last  place,  consider  Prob.  VII.     Here  we  may  write 

U—   I   'yds,  so  that  v  =  y,  V:^  —  o,  Vy  —  i,  and  (25)  gives 

t/So 

-  = and   r  —  —  y  sec  B.      Hence,  in   this  case,  the 

r  y 

radius  of  curvature  must  equal  the  normal  estimated  in  an 

opposite  direction,  and  this  is   known    to  indicate   that  the 

curve  is  a  catenary,  the  directrix  being  the  axis  of  x. 

283.  In  all  these  problems  we  shall  obtain  the  same  equa- 
tions at  the  limits  for  the  determination  of  the  arbitrary  con- 
stants as  we  would  if  we  had  regarded  x  as  the  independent 
variable.  For  suppose,  first,  the  curve  is  to  connect  two 
fixed  points.     Then,  as  shown  in  Art.  279,  the  hmiting  terms 


34^  CALCULUS  OF  VARLATIONS. 

Will  take  the  form  of  (19),  and  v  and  /  being  always  equal,  they 
will  entirely  vanish,  so  that  the  constants  must  be  determined 
by  the  circumstance  that  the  curve  is  to  pass  through  two 
fixed  points,  which  are  evidently  the  same  conditions  as  we 
would  have  obtained  had  we  assumed  x  as  the  independent 
variable.  If  we  next  require  that  the  curve  shall  always  have 
its  extremities  upon  two  fixed  curves  whose  equations  are  as 
in  Art.  278,  then  we  shall  obtain  equations  (21).  Now  the  first 
of  these  equations  gives  no  direct  condition  regarding  the 
limits,  but,  with  the  aid  of  the  second,  serves  merely  to  deter- 
mine ^  in  (16),  <;  being  an  additional  constant  necessarily  intro- 
duced by  the  employment  of  the  new  quantity  /.  But  divid- 
ing the  second  of  these  equations  by  j/,  and  multiplying  by 
\yx)v  we  find  (i  -^  f'yo^^  —  o,  and  a  similar  equation  for  the 
lower  limit.  These  equations  show  that  the  required  curve 
must  meet  its  limiting  curves  at  right  angles,  which  conditions 
are  also  the  same  as  would  have  been  obtained  had  we  assumed 
X  as  the  independent  variable. 


Problem  XLVI. 

284.  Lef  V  and  u  be  any  functions  of  x  and y  only,  with  con- 
stants, and  let  it  be  required  to  jnaximize  and  minimize  the  expres- 
sion 

Here,  as  before,  because  s  has  been  made  the  independent 
variable,  x  and  y,  and  consequently  their  variations,  cannot 
be  regarded  as  entirely  independent.  But  equation  (6),  Art. 
273,  must  always  hold  between  x  and  j/;  and  as  this  gives  an 
imphcit  relation  between  them,  the  variation  of  that  equation 
must  involve  such  a  relation  between  their  variations.  Hence, 
multiplying  the  variation  of  (6),  as  before,  by  an  unknown 


OTHER  METHODS  OF   VARIATIONS.  347 

quantity  /,  and  transforming  the  variations  by  equations  (8)  and 
(lo),  we  may,  as  before,  write  equation  (ii).  Art.  274. 

Now  it  will  appear,  by  reasoning  precisely  like  that  em- 
ployed in  the  last  problem,  that  to  vary  ^in  the  most  general 
manner,  even  when  the  required  curve  is  to  pass  through  t\\  o 
fixed  points,  we  must  add  to  the  terms  at  the  limits  the  terms 
V^ds^  —  V^ds^.  For  it  is  evident  that  the  reasoning  there  used 
would  be  equally  applicable  if,  instead  of  supposing  £/  to  be  a 
function  of  x  and  y  only,  it  had  been  any  function  of  x,  y,  x', 
^''\y,y'j  etc.  Now  varying  (i),  adding  equation  (11),  Art.  274, 
and  integrating  by  parts  as  usual,  we  shall  obtain 

6U-^l\  ds,  -  V,  ds^  +  {u  +  lx'\  6x, 

—  {u  +  lx'\  dx,  +  /  J//'  dy^  —  l.y^Sy^ 

+  r^  \  \yx  +  u^x'  -  u'-  {ix'y^Sx  +  {_vy  -f  uyx'-  {iyy^dy\ds 

=  L,-L,  +£'\MSx  +  NSy\ds.  ■  (3) 

Here,  as  before,  L^  —  L^  and  the  integral  must  severally  van- 
ish whatever  be  the  value  of  /. 

If  now,  as  before,  we  suppose  /  to  be  such  a  quantity  as 
will  reduce  M  or  N  to  zero  throughout  U,  it  will  appear  by 
the  same  reasoning  as  before  that  the  other  must  vanish  also. 
Making  M  and  N  zero  in  (3),  we  have 

(ix'  -\-uY  =  v^-\-  u^x',      {lyy  =  Vy  +  tiyx\         (4) 

Multiplying  these  equations  respectively  by  x^  and  y,  and 
adding,  observing  equation  (6),  Art.  273,  we  have 

I'  -\-  u'x'  =  v^x'  -|-  Vyy'  -\-  x'(uxx'  -\-  Uyy')  =  v'  -\-  u' x' .      (5) 

Hence,  as  before, 

l'  =  v'        and     l=v-^c.  (6) 


34^  CALCULUS  OF  VARIATIONS. 

285.  Now  in  determining  c  we  must  remember,  as  before, 
that  if  we  can  express  L^  —  L^  in  terms  of  ds^  and  ds^,  we  may, 
since  these  quantities  are  independent,  equate  their  coefficients 
severally  to  zero ;  so  that  we  need  here  consider  but  one  limit. 
Let  us  first  suppose  that  the  curve  is  to  pass  through  two.  fixed 
points.  Then,  taking  the  value  of  L^  from  (3),  and  substitut- 
ing in  it  the  values  of  dx^  and  dy^  from  equations  (17),  Art.  277, 
and  remembering  equation  (6),  Art.  273,  we  find  v^  —  l^  =  o, 
which  shows,  as  before,  that  v  =  I  throughout  U,  c  in  (6) 
being  zero. 

Next  suppose  the  curve  is  to  connect  two  fixed  curves 
whose  equations  are  as  in  Art.  278.  Then  in  L^  substitute  the 
value  of  (^F,  found  by  transposing  equation  (18),  Art.  278,  and 
equate  the  coefficients  of  ds^  and  Sx^  severally  to  zero,  because 
these  quantities  must  be  independent.     Then,  we  shall  have 

«. + /,-^-,' + hyj:  =  o,    V, + u,x: + /j'.'^//.'- /,^." = o.  (7) 

Multiplying  the  first  of  these  equations  by  x^  and  subtracting 
the  second  from  the  product,  we  shall,  by  observing  equation 
(6),  Art.  273,  have  l^  —  i\  =  o,  so  that  here  also  v  =^  I. 

286.  Putting  v  for  /,  and  differentiating  the  first  term  in 
each,  equations  (4)  become 

Vx  —  v'x'  -\-  Uxx'  —  u'  ^=  vx'\         Vy  —  v'y'  -\-  iiyx'  =.  vy" .     (8) 

Now  multiply  the  first  term  in  each  of  these  equations  by 
x''  ^  y,  and  put  for  v'  and  u'  their  values.  Then  factoring, 
we  have 

y^Oxy'  —  i^yx'  —  u,}  =  vx",        x'ivyx'  —  v^y'  +  u,])  —  vy" .     (9) 

Multiplying  the  first  of  these  equations  by  y\  the  second  by 
x\  and  subtracting  the  second  from  the  first,  we  readily  ob- 
tain, as  before, 

-—  —  -  (z'x  cos  A  -{-  Vy  cos  B  +  ^y)'  (10) 


OTHER  METHODS  OF  VARIATIONS.  349 

287.  Let  us  next  apply  this  formula  to  a  few  cases,  begin- 
ning with  Prob.  XV.  Here  U  —J^  {yji:' -\- a)  ds,  so  that  v  =  a, 
Vx  =  o,  Vy  =  o,  u  =f,  Uy=  I.     Therefore  equation  (lo)  gives 

-  z= .     Hence  the  curve  must  be  a  circle,  since  r  is  a  con- 

r  a 

stant.  The  negative  sign  is  in  this  case  as  it  should  be,  be- 
cause it  has  been  shown  that  a  must  be  negative. 

Turn  next  to  Prob.  XVI.     Here  U  =fj\/x'  +  ay)ds ;  so 

that  V  =  ay,  v^  —  o,  Vy  =  a,  it  =  y,  Uy  —  2y ;  and  equation  (lo) 
gives  • 

I  _       cosB      2 
r  y  a 

But =  -,  n  being  the  normal;  and  as  we  have  already 

y  n 

shown  that  a  must  be  negative,  we  may  write  — | —  =  — -. 

r      71       A 

We  cannot  in  this  case  proceed  to  the  solution  obtained  in 
Prob.  XVI.  without  expressing  the  value  of  r  and  integrating 
as  in  that  problem,  although  it  is  evident  enough  that  the 
sphere  will  satisfy  the  last  equation. 

We  may  remark,  in  passing,  that  Probs.  XVII.  and  XVIII. 
are  to  be  regarded  as  belonging  to  the  preceding  problem, 
because  the  factor  of  ds  is  a  function  of  x  and  y  only,  together 
with  constants. 

288.  Here  also  the  conditions  for  the  determination  of  the 
two  constants  which  will  enter  the  complete  integral  of  equa- 
tion (id)  will  be  always  the  same  as  though  we  had  assumed 
X  as  the  independent  variable.  For  if  the  curve  must  pass 
through  two  fixed  points,  we  shall  have  for  the  upper  limit 

L,  =  {v,  —  /,)as,  =  (v,  —  v,)ds,. 


350  CALCULUS  OF  VARLATIONS. 

That  is,  the  Hmiting  terms  will  vanish  as  they  would  by  the 
other  method.  But  suppose  the  curve  is  to  connect  two 
fixed  curves.  Then  if  x  were  the  independent  variable,  we 
would  obtain  for  the  upper  limit 

«Xi+/V+'''.(i+^///)  =  o; 

and  multiplying  by  x\  remembering  equation  (6),  Art.  273, 
we  shall  obtain  the  first  of  equations  (7).  Now  the  second  of 
these  equations  gives  no  new  condition,  but  merely  enables 
us  to  determine  the  constant  c  in  (6).  To  ascertain  these  con- 
ditions, let  0  be  the  angle  between  the  required  and  the  upper 
fixed  curve  at  their  intersection,  t  the  angle  whose  tangent  is 
f\  and  a  the  angle  whose  cosine  is  x' .  Then,  multiplying  the 
first  of  equations  (7)  by  cos  /,  we  have 

u^co^  t  -f-  ^i(cos  a  cos  t  -f-  sin  a  sin  f)  = 

u^cos  t  -\-  v^co^  0  —  o.  (i  i) 

Problem  XLVII. 

289,  Let  r  be  the  radius  of  curvature  of  a  plane  curve ^  and  V 
any  function  of  r  and  constants.  Then  it  is  required  to  determine 
the  conditions  which  will  maximize  or  minimize  the  expression 


Here 


u=fyds.  (I) 

SU=V,ds,-  V, ds,  +/'' VrSrds.  (2) 


Now  the  following  equations  are  known  to  be  true : 

-^R=  y'x"  -  x'y",         \  =  R'  =  x"'  ^  y"\ 
r  r 


\    (3) 


x" +/^  =  I,       x'6x'  +ysy  =  o,      x'x''  +yy'=  o.  J 

We  must  now  obtain  dr.     We  have 

■  s{R')  =  2{x"dx"+y'dy')=^  ^=^-. 


OTHER   METHODS   OF    VARIATIONS.  351 

Whence  dr  ^  -  r\x"dx"  -^y"^y"\  (4) 

Hence,  proceeding  as  before,  and  putting  v  for  F^r^  we  have 
dU=  V,ds,-  V,ds, 
Ar  r\-  v{x"dx"-^y"Sy")  +  l{x'dx'  ^  y'dy')\ds  =  o.     (5) 

Whence,  as  usual,  we  obtain,  after  changing  signs,  the  equa- 
tions 

iyx")"  +  {Ix')'  =  O,         {vy")"  +  (//)'  =  o,  (6) 

and 

(.^y  y  j^l^'  ^a  =  vx'"  +  v'x"  +  lx\  I 

Multiplying  the  first  of  these  equations  by  y' ,  the  second  by 
x' ^  and  subtracting  the  second  from  the  first,  we  have 

yi^yx'"  -  x'y'")  +  v\y'x"  -  x'y")  =  ay'  -  bx'  =  vR'  +  Rv' .  (8) 

Whence  „       ^^   2  ,      ,  /  x 

vR  =  Vrr^  =  ay  —  bx  -^  c.  (9) 

290.  It  will  be  seen  that  in  this  case  /  has  been  eliminated, 
and  we  will  now  examine  the  method  of  determining  the  con- 
stants in  (9).  Consider  the  terms  at  the  upper  limit,  arising 
from  the  usual  transformation  of  (5).     These  are 

VJs,+  \(vx'y-\-lx'\,Sx, 

+  \{vyy  +  ly'\.  Sy,  -  v,{x"dx'  +y'Sy\  =  o.        (10) 

Now  it  at  once  appears  from  (7)  that  the  coefficients  of  ^x^ 
and  Sj/^  are  respectively  a  and  b ;  and  if  for  6/  we  put  its  value 

derived  from  the  fourth  of  equations  (3),  the  terms 

y 

beyond  dy^  will  become 


352  CALCULUS   OF  VARIATLONS. 

Hence  the  terms  at  the  upper  limit  become 

VJs,-\-adx,^bdy,  -  I  ^-  I  dx:  =  o;  (12) 

and  a  similar  equation  will  evidently  hold  at  the  lower  limit. 

Now  the  last  term  of  the  first  member  of  (12)  is  evidently 
independent  of  the  others,  so  that  we  must  have  Vrr"  =  o  at 
both  limits.  Now  suppose  the  line  joining  the  extremities  of 
the  required  curve  be  assumed  as  the  axis  of  x.  Then,  because 
y  and  F^r^  vanish  at  both  limits,  we  have,  from  (9), 

o  =1  —  bx^A^  c        and     o  —  —  bx^-\-  c\ 

so  that  b  and  c  must  vanish,  and  then  (9)  becomes 

VrT'^ay,  (13) 

291.  Suppose  the  curve  is  to  pass  through  two  fixed  points. 
Then  the  terms  at  the  upper  limit  become 

V^  ds^  -\-  aSx^  =  (v  —  <^^0i  ^^1  —  ^' 

the  second  member  resulting  from  the  elimination  of  dx^  by 
means  of  equation  (17),  Art.  277 ;  and  a  similar  equation  holds 
for  the  lower  limit. 

But  suppose  the  curve  is  to  connect  two  fixed  curves 
whose  equations  are  as  heretofore.  Then  the  terms  at  the 
upper  limit  are 

Fj  ds^  -["  ^^^1  +  ^^Ji  —  ^'  (14) 

Ehminating  dj,  by  means  of  equation  (18),  Art.  278,  and  then 
equating  severally  to  zero  the  coefficients  of  ds^  and  Sx^,  we 
shall  obtain 

K  +  ¥/<  -  hy!  =  0,         a  +  bf:  =  0;  (15) 

and  similar  equations  for  the  other  limit.  Now  if  the  axis  of 
X  join  the  points  of  intersection  of  the  required  curve  and  the 


OTHER  METHODS  OF    VARIATIONS.  353 

two  fixed  curves,  b  will  vanish,  while  a  cannot,  as  appears 
from  equation  (13) ;  so  that  the  second  of  equations  (15)  can 
only  be  satisfied  by  supposing//  to  be  infinite. 

Hence  the  tangents  to  the  two  fixed  curves  at  their  points 
of  intersection  with  the  required  curve  must  be  at  right 
angles  to  the  line  joining  those  points. 

292.   As  an  example  of  the  foregoing  theory,  consider 
Prob.  III. 
Here 


SO  that  V=r,  [>=  i,  and  equation  (13)  gives  r"  =  ay.  Now 
as  the  axis  of  x  in  this  case  joins  the  two  extremities  of  the 
required  curve,  it  is  readily  seen  that  the  cycloid  having  its 
cusps  upon  the  axis  of  ;r  is  a  solution,  because  in  such  a  cycloid 
r  =  2  VDy,  D  being  the  diameter  of  the  generating  circle. 

293.  Another  interesting  apphcation  is  the  following: 

An  elastic  spring  AB  is  adjusted  between  two  right  lines  so 
as  to  be  tangent  to  both  at  its  extremities  A  and  B ;  it  is  required 
to  determine  the  form  which  the  spring  must  assume  in  order  to 
be  in  equilibrium. 


According  to  the  principle  of  Daniel   Bernoulli,  the  curve 
AB  must  be  such  as  to  minimize  the  expression  U  =J     --. 

J  2 

Hence  V  =  -,  V,.  —  — r->  and  equation  (9)  becomes 

r  r 

=  ay  —  bx  -\-  c,  (16) 


354  CALCULUS  OF  VARLATLONS. 

But  since  AB  is  compelled  to  be  tangent  to  the  lines  AC  and 
BD,  its  extreme  tangents  have  a  fixed  inclination  to  the  axis  of 
X,  and  therefore  d;r/,  dj/,  ^^J  and  Sj/J  vanish,  and  we  need 
not  now  have  Vrr""  =  o  at  either  limit.  But  equations  (15)  are 
universally  true,  and  the  second  of  these  gives 

a-^bf  —  o        and     a  +  bF'=^  o.  (17) 

But  since  the  lines  A  C  and  BD  are  not  parallel,  the  constants 
/'  and  F' ,  which  are  the  tangents  of  the  inclinations  of  these 
lines  to  the  axis  of  x,  are  unequal ;  so  that  in  this  case  we  find 
that  <^  and  b  must  vanish.  Then,  by  (16),  we  find  that  r  is  a 
constant,  so  that  AB  must  be  a  circular  arc  if  r  be  finite. 

But  now  the  first  of  equations  (15)  would  appear  to  give 

V  —  -2=0  for  both  limits ;  which  evidently  cannot  be  true. 

To  obviate  this  difficulty  we  must  suppose  the  spring  to  have 
a  given  length.  Then  ds^  and  ds^  will  vanish,  and  the  first  of 
equations  (15)  will  not  necessarily  hold. 

But  under  this  supposition  we  should,  according  to  Euler's 

method,  have  written  V=  —  -{-  d,  which  would  produce  no 

change  in  any  equation  except  the  first  of  equations  (15) ;  and 

this,  when  a  and  b  vanish,  would  give- +  <^'=  o  at  either  limit, 

which  presents  no  difficulty. 

T/iird  Method. 

294.  We  have  already  seen  that  when  x  is  the  indepen- 
dent variable,  we  are,  although  the  supposition  is  unnatural, 
permitted  to  vary  x ;  and  in  like  manner,  when  s  is  the  inde- 
pendent variable,  we  may  ascribe  variations  to  s  throughout 
the  range  of  integration.  Indeed,  this  is  the  method  usually 
adopted ;  and  as  we  are  generally  obliged  to  increase  or  de- 
crease s  at  its  limits,  the  method  does  not  seem  altogether 


OTHER  METHODS  OF   VARIATIONS.  355 

unnatural.     The  following  illustration  may  perhaps  aid  us  in 
forming  a  better  conception  of  the  two  methods. 

295.  Suppose  we  had  a  curve  AB  connecting  two  fixed 
points  or  two  fixed  curves,  and  suppose  the  curve  to  be  formed 
of  non-elastic  wire  on  which  notches  are  placed  at  our  plea- 
sure, the  wire  extending  somewhat  beyond  A  and  B.  Then 
when  we  vary  the  form  of  AB  in  the  most  general  manner 
consistent  with  variations,  we  shall,  in  general,  find  that  we 
are  unable  to  make  the  new  curve  connect  the  two  points  or 
curves  without  either  adding  or  excluding  certain  wire  ad- 
jacent to  A  and  B.  Still  the  distance  of  any  notch  from  some 
given  notch — that  is,  s — undergoes  no  change,  a  positive  or 
negative  increment  merely  being  added  to  the  limits.  This 
may  illustrate  what  takes  place  in  the  first  method. 

Now  suppose  the  original  piece  to  be  expanded  by  heat  or 
contracted  by  cold  until  it  is  able  to  form  the  required  arc  of 
the  derived  curve.  Then,  although  we  increase  or  diminish 
the  length  of  the  arc  AB,  we  do  not  add  or  exclude  any  wire. 
But  now  the  distance  of  any  notch  from  the  given  notch,  or  s, 
will  have  undergone  an  infinitesimal  change  ;  that  is,  will  have 
become  s^  ds.  But,  to  render  the  illustration  complete,  we 
must  suppose  the  motion  of  any  particular  notch  to  be  capable 
of  taking  either  a  positive  or  negative  direction,  or  of  becom- 
ing zero,  or,  in  short,  of  following  any  law  we  please.  In  this 
'case  we  would  have  an  illustration  of  the  method  which  we 
are  now  about  to  employ. 

296.  Let  us  now  examine  the  mode  of  employing  this 
method. 

Assume  the  equation 

U^£vds,  (.) 

where   V  is  any  function  of  s,  x,  x' ,  x'\  .  .  .  .  y,  y',  y"  .  .  .  . 
Now  when  we  vary  s,  x,  y,  etc.,  the  reasoning  in  the  begin- 


356  CALCULUS   OF  VARIATLONS. 

ning  of  Art.  264  is  rendered  applicable  to  the  present  case 
by  reading  s  for  x.  Moreover,  all  the  equations,  including 
(6),  will  be  true  if  for  x  we  substitute  s  in  the  limits,  the  differ- 
entials and  the  variations.     Beginning  then  with  (6),  we  have 


But 


t/so      ds  ^so  ^  ' 


where  accents  denote  total  differential  coefficients,  while  literal 
suffixes  will  denote  partial  differential  coefficients ;  so  that 

V'=V,-^  V^x'  +  V^,x"  +  V^.x"'  +  etc. 

+    Vyy    +     Vy.f  +     Vy.>y"'   +    CtC.  (4) 

Now,  to  the  first  order,  we  have 

6V^  V,ds-\-  VJx+  V^.^x'+  V^nSx" -\-Qic. 

+  Vy^y  +  Vy>dy  +  Vy..dy"  +  etc.  (5) 

Hence 

+  r  \  Vx^^  +  ^0"^^'  +  Va:'>dx"  +  etc. 

+  Vy^y + Vy,sy^  Vy,.s/'+  etc. 

-(  F^^'+  V^,x"  +  V^,.x"'  +  etc. 

+  Vyy'  +  Vy,y"  +  F,„/"'  +  etc.)  Ss\ds.     (6) 
Now  employing  gd  as  before  (Art.  265),  let 

GD^  =z  (^;ir  —  x'ds         and     cb?^  =  6y  —  y'Ss. 


OTHER   METHODS  OF   VARIATIONS,  35/ 

Then,  by  the  same  method  as  that  employed  in  Art.  265,  we 
obtain 

Sx'  =  {G^y  +  x"Ss,         dx''  =  {G^y  +  x'^'Ss,         etc.,  ) 

I     (7) 
d/  =  {ooyy  +y'^s,          dy"  =  {Goyy^  +/''^s,  etc.   ) 

But  these  equations  are  of  course,  Kke  those  in  Art.  265, 
true  to  the  first  order  only.  By  the  use  of  these  equations, 
(6)  becomes 

+  pi  V,co-  +  VAoo^y  +  V,.{c^r  +  etc. 

+  VyGDV  +  Vy^Goyy  +  VyioDvy  +  etc.  \  ds.      (8) 

Hence,  by  the  usual  transformation,  and  giving  for  brevity 
only  the  general  form  of  the  terms  at  the  limits,  we  have 

6U=^  Vds-\-{V^>-  F^,/  +  etc.)c^+(r^.,-etc.)(o^y  +  etc. 

+  {Vy'-  Vy>/  +  etc.)c^^  +  {Vy.  -etc.)(G.^y  +  etc.    . 

+  {Vy-  Vy/  +  Vy>> "  -  ^tc:)ooy\ds.      (9) 

297.  But  dx  and  Sy,  and  consequently  oof^  and  coy,  are  not 
wholly  independent,  because,  whether  we  vary  s  or  not,  the 
equations 

x'""  J^y:=i         and     x'x"  +  y'y"  =  0  (lo) 

must  always  hold  throughout  both  the  primitive  and  derived 
curve.  If,  therefore,  we  wish  to  maximize  or  minimize  U,  and 
for  this  purpose  equate  (^t/to  zero,  we  must,  as  before,  in  order 


358  CALCULUS  OF  VARIATIONS. 

to  obtain  any  available  equations  of  condition,  employ  the 
method  of  Lagrange.     Now  from  (lo)  we  have 

x'dx'  +y^/  =  0  =  x\g^)'  +  x'x"Ss-^y\Goy)'  ^y'y"Ss 

=  x\oo^)'  -\-y\ojy)'  4-  {x'x"  +yy')^s  =  x'{GD^y  -\-y{oDyy.  (i  i) 

Therefore,  /  being  an  undetermined  quantity,  we  may,  as  be- 
fore, write 


£'i\x'{c^)'^y{oovy\ds 


Now  transform  this  equation  and  add  it  to  (9),  and  let  L  de- 
note the  general  form  of  the  limiting  terms  L^  —  Z^,  M  and  N 
being  the  respective  coefficients  of  ooP^ds  and  coyds  under  the 
integral  sign.     Then  we  shall  have 

L^Vds-\-{  V^^  -  V^>/+  etc.  +  Ix')  G^  +  ( V^u-  etc.)  (c^)'+  etc. 

+  {Vv'-  yy"'+  etc.  +  ly)  ^y+  ( Vy.  -  etc.)  {oovy  +  etc.,   (12) 

J/=  F.  -  Vy  +  V,."  -  etc.  -  {IxJ,  (13) 

N=Vy-  vy  +  Vy."  -  etc.  -  {lyy.  (14) 

Now  it  is  evident  that  (13)  and  (14)  are  the  same  differen- 
tial equations  as  we  would  have  obtained  had  we  followed 
the  preceding  method,  and  ascribed  no  variation  to  s,  I  of 
course  in  each  case  being  supposed  to  be  so  taken  as  to  cause 
either  M  or  N  to  vanish,  so  that  the  other  will  vanish  also. 
Hence,  since  the  general  solution  will  have  the  same  form  as 
before,  it  will  be  necessary,  in  further  comparing  the  two 
methods,  to  consider  only  the  terms  at  the  limits. 

298.  It  may  be  observed,  in  the  first  place,  that  the  gen- 
eral form  of  the  limiting  terms  is  the  same  by  the  two  methods ; 
6s^,  8s^  and  the  cos  and  their  differential  coefficients  in  the 


OTHER  METHODS  OF   VARIATIONS.  359 

second  method  replacing  ds^,  ds^  and  the  (^'s  in  the  first.  It 
would  appear,  therefore,  that  we  might  safely  assume  that  the 
same  conditions  at  the  limits  could  be  ultimately  obtained  by 
the  two  methods.  But  as  it  has  not  been  deemed  necessary 
to  consider  the  most  general  form  of  V  by  the  other  method, 
it  will,  we  presume,  be  sufficient  to  give  Fthe  same  degree  of 
generality  in  this  ;  that  is,  to  show  that  in  the  three  preceding 
problems  the  same  equations  at  the  limits  are  obtained  by 
either  method.. 

Suppose  we  make  V  a  function  of  x  and  y  only ;  that  is, 
apply  this  method  to  Prob.  XLV.  Then,  by  (12),  we  have,  for 
the  upper  limit, 

L,  =  Vfis,  +  {lx'c^\  +  (ly'c^y\  =  0.  (15) 

Now  suppose  the  curve  is  to  pass  through  two  fixed  points. 
Then  dx\  and  S)\  vanish,  because  by  this  method  x^  and  y^ 
mean  the  co-ordinates  of  the  actual  extremities  of  the  arc,  al- 
though 6x^  need  not  vanish,  as  the  arc  may  have  undergone  an 
alteration  in  length.  Hence  (0/^)^=  —  x^Ss^,  {oofii)^—  —  yl^s^, 
and  (15)  gives 

Z,^|F-/(y^+yOh-o;  (16) 

so  that  Fj  =  /,.  ^ 

Next  suppose  the  curve  is  to  connect  two  fixed  curves  whose 
equations  are  as  usual.  In  this  case  we  shall  have  Sy^  =//  ^x^. 
Substituting  this  value  in  (15)  and  equating  severally  to  zero 
the  coefficients  of  ^s^  and  Sx^,  because  these  quantities  are 
entirely  independent,  that  of  Ss^  will  give  the  second  and  third 
members  of  (i6),  while  that  of  dx^  will  give 

(/y  +  //'/X  =  o. 

This  is  the  same  as  the  second  of  equations  (21),  Art.  279,  the 
interpretation  of  which  is  given  in  Art.  283. 


360  CALCULUS  OF    VARLATIONS. 

299.  Next  consider  Prob.  XLVI.  Here  V  —  v  -\-  ux\ 
V  and  u  being  functions  of  x  and  y  only,  so  that  V^'  =  u. 
Therefore  (12)  gives 

{v  +  ux'),ds,  +  {u  +  lx'\{p^\  +  hy;{oDy\  =  o.  (17) 

If  now  the  curve  is  to  pass  through  two  fixed  points,  ^x^  and 
<Sj/^  will  vanish,  and  putting  for  go^  and  aoy  their  values,  the  co- 
efficient of  6s^  will  take  the  form  of  the  second  member  of 
(16),  which  shows,  as  before,  that  V  =  I. 

Next  suppose  the  curve  is  to  connect  two  fixed  curves. 
Then  we  have  dy^  —  f^dx^.  Now  substitute  in  (17)  the  values 
of  00^  and  Qoy,  and  eliminate  Sy^.  Then,  as  ds^  and  Sx^  are  in- 
dependent, we  must  equate  their  coefficients  severally  to  zero. 
That  of  Ss^  will,  as  before,  assume  the  form  given  in  (16),  show- 
ing that  V—  I,  while  that  of  dx^  will  become 

{u  +  /x^+/yr\=-o. 

But  this  is  the  first  of  equations  (7),  Art.  285,  which  has  been 
already  considered  in  Art.  288. 

300.  In  the  last  place,  consider  Prob.  XLVII.  Here  we 
have 

=  KSs-  V,^s,+£\-  V^ds+dV)ds,  (I) 

Now 

v  =  vy,  (2) 

But  employing  the  reasoning  by  which  we  obtained  Sr  in  Art. 
289,  only  putting  for  every  d  an  accent,  we  find 

/=-rW" +/'/");  (3) 

and  therefore,  putting  z'  =  Vrr\  we  have 

v':=  -  v{x''x"' + yy^).  (4) 


OTHER  METHODS  OF  VARIATIONS.  36 1 

We  also  have 

6  V=  VrSr  =  -  v{x"6x"  +  y"^y"\  (5) 

dr  having  the  same  value  as  in  equation  (4),  Art.  289,  although 
dx"  and  Sy"  have  not  now  the  same  values.  Therefore,  by 
substitution,  (i)  becomes 


+X^ 


v{x"dx"^y"dy")  +  v{x"x"'  •\-y"y"')ds\ds 

=  vM  -  v,ds,  +£'{-  ^[{c^r + i^^r]  w^'      (6) 

Hence,  integrating  and  employing  the  method  of  Lagrange, 
we  shall  evidently  obtain  for  the  general  solution  the  same 
differential  equations  as  before  (equations  (6),  Art.  289). 
Now  the  terms  at  the  upper  limit  will  be 

-v,{x"{<^y+y'{wyY\,  =  o,  (7) 

which  is  similar  to  equation  (10),  Art.  290. 

But  from  equation  (11),  Art.  297,  we  see  that  we  can  elimi- 
nate {coyy  in  the  same  manner  as  we  did  dy'  in  Art.  290 ;  and 
as  equations  (7),  Art.  289,  are  obtained  in  the  general  solution, 
the  terms  at  the  limit  will  become 

V^^s,  +  a{c^l  +  d{Goy\  -  ( Vr  r\  (c^)/  =  o.  (8) 

But  since  (ce?^)/  =:  Sx^'  —  x^''ds^,  if  Sx^'  be  unrestricted,  so  will 
(g>^)/,  and  its  coefficient  must  vanish  ;  so  that,  as  before,  b  and 
c  become  zero,  and  we  have 

VM  +  a{c^\  =  o.  (9) 


362  CALCULUS  OF  VARLATIONS. 

Then  if  the  extremities  be  fixed,  d;ir, becomes  zero,  and  we 
have,  as  before  (Art.  291),  (F—  ax')^  =  o.  But  if  the  extremi- 
ties are  to  be  upon  two  given  curves,  then  the  terms  at  the 
limits  become 

V,6s,  +  a{GD^),  +  d{c<oy),  =  o.  (10) 

Now  substitute  in  (10)  the  values  of  {go^)^  and  (00^)^,  and  also 
for  Sj\  the  value  f^^x^.  Then  equating  severally  to  zero  the 
coefficients  of  Ss^  and  ^x^^  we  shall  have 

Fi  —  axl  —  by  I  =  o         and     a  +  d//  =  o.  (i  i) 

Eliminating  a  from  the  first  of  these  equations  by  means  of 
the  second,  it  becomes 

K  +  ¥/<  -  h^  =  o. 

But  the  last  two  equations  are  equations  (15),  Art.  291,  and 
we  have,  therefore,  the  same  conditions  as  formerly. 

Thus  we  see  that  while  the  equations  for  the  general  solu- 
tion given  by  the  two  methods  are  always  necessarily  the 
same,  the  limiting  equations  are  also  the  same  eventually,  at 
least  so  far  as  we  have  carried  our  investigations. 


CHAPTER  II. 

MAXIMA  AND   MINIMA  OF  SINGLE   INTEGRALS   INVOLVING  TWO 
OR   MORE   DEPENDENT   VARIABLES. 


Section  I. 

CASE    IN    WHICH   THE    VARIATIONS    ARE    UNCONNECTED  BY 

ANY  EQUATION 

Problem  XLVIII. 

301.  It  is  required  to  determine  the  curve  of  mnimum  length 
which  can  be  drazvn  between  two  fixed  points  given  at  pleasure  in 
space. 

Let  ds  be  an  element  of  the  required  curve.  Then  since 
the  curve  is  to  be  situated  in  space,  and  is  no  longer  neces- 
sarily plane,  we  have 


ds  =  Vdx'+  d/+  dz"  =:|/l  +  ^  -f  ^  ^;r  =  |/l+y^+^'V;ir. 
Therefore  the  expression  to  be  minimized  in  this  case  is 

Now  it  is  evident  that  here,  as  in  Prob.  I.,  we  must  compare 
the  required  curve  with  such  as  are  drawn  indefinitely  near 
at  every  point ;  and  it  is  also  evident  that  by  giving  to  y'  and 


3^4  CALCULUS  OF  VARLATIONS. 

z'  indefinitely  small  variations,  these  variations  being  wholly 
unrestricted  as  to  sign,  we  can  make  any  infinitesimal  change 
we  please  in  the  form  of  the  primitive  or  required  curve. 
Now  if  we  change  y'  into  y'  -|-  dy\  and  <2d  into  z'  -\-  Sz' ,  while 
X  undergoes  no  change,  the  corresponding  alteration  in  the 
length  of  the  required  curve  will  h^  dU\  and  the  method  ot 
finding  SU  vn  its  untransformed  state  needs  no  explanation  ; 
that  already  given  being  perfectly  general  whatever  be  the 
quantities  involved  in  U, 

302.   Therefore,  to  the  second  order,  we  have 

dU=  r^  \  -—^jL—:r^. 6y'  +  --      ^'  8z'  \ dx 

+  Lrj__I_-Ml_„.y.+         '+/'_-..>» 

Now  it  needs  no  additional  explanation  to  show  that  if  U  is 
to  become  a  minimum,  the  first  integral  in  (2)  must  vanish, 
while  the  second  must  become  invariably  positive.  Hence, 
to  the  first  order,  we  have 

6U=r'\  y'  Sv'\.  ^' Sz\dx 

=  r'   , ^^  Sy'dx  +  r^  ^' dz'dx  =  o.  (3) 

But  since  z  is  also  a  function  of  x,  we  may  put  z,  z\  z",  etc., 

for  y,  /,  y\  etc.,  in  the  reasoning  of  Art.  9.     Then  we  shall 

d'^'^z 
find  (^^^)  =:  -^— -.     In  like  manner  it  is  evident  that  when  x 


SHORTEST  CURVE  IN  SPACE.  365 

receives  no  variation,  if  we  had  any  number  of  variables  y,  z, 
u,  etc.,  all  regarded  as  functions  of  x,  the  reasoning  of  Art.  9 
would  apply  to  each,  and  we  would  have 

dy.,=  ^^,         tf^«)=f^-,         <y««^^'i.     etc.;     (4) 
-^  dx""  dx'^  dx''  ^^^ 

and  these  equations  w^ill  hold  whether  y,  z,  w,  etc.,  are  inde- 
pendent, or  are  connected  by  some  equation. 

303.    Therefore,  transforming  SU'vcv  the  usual  manner,  we 
have 

6U=  \  y'  X  Sy^  -  \      __Z____-j  6y, 


—  I  —       -^  ^  Sydx  —  I -  Sz  dx  =  o 

^^0  dx  \/' I  j^ y^  j^  z'^  -^  ^^0  dx  4/x-[-j/'^-|-y^ 

=  K8y^-Kdy,-\.H,dz,  -  H,Sz,  ^£^\M6f^NSz\dx,  (5) 

where  J/ and  iVare,  as  previously,  total  differential  coefficients. 
But  as  the  required  curve  is  to  pass  through  two  fixed  points, 
dy  and  Sz  must  vanish  at  both  limits,  so  that  ^C/ will  consist 
only  of  the  terms  under  the  integral  sign  in  (5). 

Now  Sy  and  Sz  are  here  entirely  independent.  For  we 
may  suppose  the  derived  curve  to  be  obtained  by  varying 
one  of  the  quantities  y  or  z,  thfe  other  undergoing  no  change 
whatever ;  or  we  may  suppose  it  to  be  such  as  would  require 
us  to  vary  both.  Hence,  that  (^f/may  vanish,  the  two  inte- 
grals in  (5)  must  severally  vanish. 

But  both  (^Kand  dz  are  entirely  in  our  power,  and  are  each 
as  unrestricted  as  is  dy  in  Prob.  I.     Therefore,  to  make  both  in- 


366  CALCULUS   OF  VARIATIONS. 

tegrals  necessarily  vanish  severally,  we  must  have  M  —  o  and 
304.  Equating  M  and  iV  severally  to  zero,  we  shall  obtain 

y 


=:  =  c  and     _       — r^—  =  c' .  (6) 

Now  solving  these  equations  by  common  algebraic  methods 
for  y'  and  z' ,  we  find  both  these  quantities  to  be  constants,  say 
a  and  a'  respectively.  Whence,  by  a  second  integration,  we 
find 

y  —  ax  -\-  b         and     z  =  a'x  -\-  b\  (7) 

the  equations  of  the  right  line  in  space. 

This  is,  of  course,  only  a  general  trial  solution,  and  to  ren- 
der it  applicable  in  any  particular  case  we  must  show,  first, 
that  real  values  can  be  obtained  for  the  four  arbitrary  con- 
stants which  it  contains,  and,  second,  that  the  terms  of  the 
second  order  in  d^^  become  positive. 

305.  Let  us  first  suppose  that  the  line  is  to  pass  through 
two  fixed  points  whose  co-ordinates  x^,  y^,  z,  and  x^,  y^,  z^ 
are  known.     Then  we  have 

and  these  equations  are  evidently  sufficient  for  the  determina- 
tion of  the  constants  a,  b,  a'  and  b' ;  and  we  see  that,  because 
these  constants  have  the  meaning  explained  in  works  on  ana- 
lytical geometry,  they  will  always  have  real  values. 

But  suppose  the  limiting  values  of  x  only  to  be  fixed  ;  that 
is,  that  the  line  is  merely  to  have  its  extremities  always  situ- 
ated in  two  fixed  planes,  each  perpendicular  to  the  axis  of  x, 
their  equations  being  x  :=^  x^  and  x  ^  x^.  Then  it  will  appear, 
by  the  same  reasoning  as  has  been  hitherto  employed,  that  the 
portion  of  (5^  f/ remaining  under  the  integral  sign  must  be  en- 


SHORTEST   CURVE   IN  SPACE.  367 

tirely  independent  of  that  which  is  free  from  this  sign.  It 
must,  moreover,  be  plain  that  the  last  statement  would  hold 
even  should  V  contain  other  dependent  variables  besides  z, 
and  will  also  hold  whether  these  variables  be  functions  of  x 
which  are  completely  independent,  or  are  in  some  manner 
connected. 

306.  Therefore,  since  L^  —  L^  must  always  vanish,  we 
must  here  have 

K^y.  -  hfy^  +  Hfiz,  -  H,Sz,  =  o.  (9) 

Now  in  the  present  case  it  is  evident  that  the  quantities  dy., 
^fof  ^^ij  ^^0  ai'e  entirely  independent,  and  hence  the  coefficients 
of  these  quantities  must  severally  vanish,  and  we  have 

h,  =  o,         //„  =  0,         H,^  o,         H,  =  o.  (10) 

But  we  see  from  (6)  that  h  —  c  and  H  =  c' ,  so  that  (7)  gives 
y  =  b  and  z  =  b',  a  and  a'  becoming  zero.  As  the  four  condi- 
tions given  by  (10)  are  here  equivalent  to  but  two,  the  con- 
stants b  and  b'  are  undetermined.  This  case  is  similar  to  that 
in  Art.  43,  the  line  being  here  also  parallel  to  x.  If  we  fix 
the  values  of  y  and  z  at  either  limit,  b  and  b'  are  determined, 
becoming  those  values  respectively ;  and  if  we  give  one  limit- 
ing value  only,  the  constant  which  equals  that  value  is  deter- 
mined, while  the  other  remains  undetermined. 

307.  It  being  possible  to  cause  the  terms  of  the  first  order 
in  ^^to  vanish,  let  us  next  consider  whether  those  of  the  sec- 
ond order  will  become  positive.  Now  it  appears  from  (5)  that 
these  terms  may  be  written 

2'J'".  (i-|_y»-^^")e 


368  CALCULUS  OF    VARIATIONS. 

and  as  we  may  regard  (i  +y +  -s''')^  as  positive,  d£/is  posi- 
tive, and  the  solution  renders  U  a  minimum. 

308.  Let  us  now  consider  the  case  in  which  the  limiting 
values  of  x  also  are  to  undergo  variation.  Here  no  new  prin- 
ciple is  involved.  For,  by  the  same  reasoning  as  before,  it 
must  be  evident  that  if  V  be  any  function  of  x,  y,  z,  u,  etc., 
and  the  differential  coefficients  of  y,  z,  u,  etc.,  with  respect  to 
X,  all  being  regarded  as  functions  of  x,  and  we  change  the 
limits  into  x^  -f-  dx^,  and  x^  -f-  dx^,  and  also  vary  all  the  quanti- 
ties except  X,  and  then  approximate  as  before  to  the  second 
order,  we  shall  merely  be  obliged  to  add  to  the  value  of  d  6^ ob- 
tained by  supposing  the  limiting  values  of  x  only  to  be  fixed, 
the  terms 

V, dx,  -  Fo dx,  +  ^  dx^  +  SV, dx,,  (i2) 

where  accents  denote  total  differentials,  so  that 

F  :=  F,  +  Vyy'  +  etc.  +  V,z'  +  etc.,  ) 

and  Hi  3) 

(^F=  VySy-\-  Vy.dy'  +  etc.  -\-V,^z-\-  V,,6z'  +  etc.  ) 

Therefore,  if  in  the  present  case  we  regard  the  terms  of  the 
first  order  only,  we  must  merely  add  to  the  limiting  terms 
already  obtained,  the  terms 


V^dx  -  VJx,  or    V(i  +y^  +  z'\  dx,  -  V{i  +/'  +  z'\dx,. 

But  it  is  plain  that  if  dx^  and  dx^  be  entirely  unrestricted, 
we  must  have  Vi  -\-  y""  -^  z""  =  o  at  both  hmits,  which  is  clearly 
impossible  without  rendering  y  or  z'  imaginary. 

309.  But  suppose  the  required  line  is  always  to  have  its 
extremities  upon  two  surfaces  whose  equations  are  known. 
Then  it  is  plain  that  the  quantities  Sy^,  dz,  and  dx^  will  not  be 
entirely  independent,  although  any  two  of  them  will  be  in- 


SHORTEST  CURVE  IN  SPACE.  369 

dependent,  so  that  if  we  can  eliminate  any  one  of  the  three, 
we  shall  have  the  same  number  of  limiting  equations  as  when 
x^  and  x^  are  fixed.  We  must,  however,  in  this  case  adopt  a 
method  somewhat  different  from  that  by  which  we  obtained 
equations  (10),  Art.  69. 

As  the  most  convenient  form,  let  the  equations  of  the  sur- 
faces at  the  upper  and  lower  limits  be  respectively 

/(.r,  y,z)^  o^f        and     Fix,  y,  z)  =  o  =  F.         (14) 

Considering  the  upper  limit,  suppose  the  required  line  when 
a  minimum  to  meet  the  surface  at  a  point  D  before,  and  at  a 
point  F  after,  having  been  varied.  Also  let  the  co-ordinates 
of  ^ — which  is,  of  course,  indefinitely  near  D — be  x^  -\-  dx^,  Y^, 
and  Z^.  Then,  when  in  f^  we  substitute  for  the  co-ordinates 
of  D  those  of  F,  we  cause  f^  to  undergo  no  change,  as  it  will 
remain  zero.  But  we  can  evidently  pass  from  D  to  Fhy  first 
passing  to  the  derived  curve  without  changing  the  value  of 
the  abscissa  x^,  and  then  tracing  along  this  curve  until  we 
reach  a  point  whose  abscissa  is  x^  -{-  dx^,  which  must,  by  the 
conditions  of  the  question,  be  the  point  F.  Now  by  the  first 
movement  we,  in  f^y  change  y  into  y  -\-  ^y,  and  ^  into  ^  -f-  S^;, 
thereby  probably  increasing  or  diminishing  /],  while  by  the 
second  we,  in  the  new  value  oi*/^,  change  x^  into  x^-\-dx^, 
which  reduces /i  again  to  zero. 

We  see,  then,  that  if  we  change  y^^  into  y^-\-  Sy^,  z^  into 
•2'i+  ^-s*!,  and  x^  into  x^  -\-  dx^,  the  increment  which  will  result 
to  f^  will  be  zero.     We  have  then,  to  the  first  order, 

{fy^y\  +  {fz^^\  +  (A  + A/  +/.  ^\dx,  =  o.  (1 5) 

This  equation  is  true  to  the  first  order  only,  since  the  complete 
increment  which  /^  Avould  receive  is  absolutely  zero,  while  we 
have  merely  obtained  that  increment  to  the  first  order.  But 
we  can  obtain  an  equation  true  to  the  second  order  by  merely 
developing  (15)  to  the  second  order,  and  equating  this  develop- 
ment to  zero. 


370  CALCULUS   OF    VARLATIONS. 

310.  To  employ  (15)  in  the  present  case  write 

X=^         and     f„^f^-.  (16) 

J  X  J  X 

Then,  observing  that  here  y' ^=^  a  and  z' =  a',  we  have,  from 

(15), 

(I  +  af,  +  a'f^,yx,+f^,dy,  -{-f,,6z,  =  o.  (17) 

We  also  have 

L,  =  V,  dx,  +  h,Sy^  +  H<5z,  ^  o. 

Substituting  the  values  of  F^,  //„  //,,  y'  and  z' ,  and  clearing 
fractions,  we  have 

(l  +  ^^  +  ^^^)^;ir,  +  ady^  +  ^'(^^,  =  o. 

Then  substituting  in  the  last  equation  the  value  of  dx^  derived 
from  (17),  clearing  fractions,  and  equating  severally  to  zero 
the  coefficients  of  6y^  and  dz^^  we  have,  after  changing  sign, 

\  (18) 

Multiplying  the  first  equation  by  a\  the  second  by  a,  and  sub- 
tracting, we  obtain 

^X.-<n  =  0..  (19) 

But,  by  reduction,  equations  (18)  become 

///I  -  ^'  -\-fm^"  —  ^^'fn  =fm  -a'  -a  {f^  -  af^^,)  =  o. 

Hence,  from  (19),  we  see  that  /^,  =  a  and  /^^,  =  a' ;  and  it  is 
clear  that  we  can  discuss  the  lower  limit  in  a  similar  manner, 
so  that 

/.  =  ^,     /n  =  ^^     ^.0  =  ^,     Fu.^^'-      (20) 


SHORTEST  CURVE  IN  SPACE.  3/1 

Now  in  determining  the  four  constants  a,  a\  b  and  b\  we 
shall  be  concerned  with  ten  unknown  quantities,  x^^  j/^,  z^,  x^, 
y^,  z^,  a,  a' ,  b  and  b' .  But  we  have,  in  addition  to  the  four 
equations  (20),  the  following  six  equations : 


/i  =  ^-^1  +  ^,         -,  ==  a'x,  +  b',         /  =:  o, 
y^  =  ax,  +  ^,         z,  —  a'x,  -\- b' ,         i^o  =  o ; 

and  it  is  evident,  therefore,  without  going  into  the  discussion 
of  any  particular  case,  that  these  ten  equations  are  sufficient 
for  the  determination  of  all  the  quantities  involved. 
Now  from  (14)  we  have,  for  the  upper  hmit, 

Udx-\-fydY^f,dZ^o, 
or 

i+/i"'+/.^'  =  o=i+«K'+«'Z';  (21) 

and  since  we  may  regard  V  and  Z'  as  belonging  to  any  right 
line  drawn  through  D,  and  also  lying  in  the  tangent  plane  to 
the  upper  limiting  surface  at  D,  the  required  curve  must  be 
normal  to  any  such  line,  and  consequently  to  the  tangent 
plane.  Therefore,  since  similar  equations  Avould  hold  for  the 
lower  limit,  we  conclude  that  the  required  straight  line  must 
be  normal  to  the  given  surfaces. 

If,  instead  of  surfaces,  the  straight  line  is  to  have  its  ex- 
tremities upon  two  curves,  let  the  equations  of  the  upper  curve 
be  dy  ^=^f'dx  and  dz  =  F'dx,  Then,  by  reasoning  like  that  in 
Art.  69,  we  shall  find 

Sy^  ■=.  {f  -  y\  dx,        and     Sz,  =  {F'  -  z'\  dx, ; 

and  recollecting  that  y  =  a,  z^  =  a\  we  shall  obtain,  by  sub- 
stituting these  values  in  the  most  general  form  of  Z,  —  L^,  the 
equation  i  -|-  a/'  -|-  a^F^  —  o,  together  with  a  similar  equation 
for  the  lower  limit,  so  that  the  line  must  be  normal  to  the  two 
limiting  curves. 


3/2  CALCULUS  OF   VARIATIONS. 

It  is  evident,  however,  that  in  these  latter  cases,  in  which 
the  limiting  values  of  x  are  not  fixed,  the  results  would  be 
sometimes  maxima  and  sometimes  minima;  and  we  must 
therefore  repeat  the  caution  frequently  given  heretofore — 
not  to  receive  as  final  any  results  obtained  by  an  examination 
of  the  terms  of  the  first  order  alone. 


Problem  XLIX. 

311.  It  is  required  to  determine  the  curve  in  free  space  down 
which  a  particle,  influenced  by  gravity  alone,  would  descend  from 
one  fixed  point,  curve,  or  surface  to  another  fixed  point,  curve,  or 
surface  hi  a  ininiimun  time. 

Assume  the  axis  of  x  vertically  downward ;  and  if  the  par- 
ticle be  supposed  to  have  an  initial  velocity  at  the  upper 
point,  which  is  the  lower  limit  of  integration,  let  h'  be  the 
height  due  to  that  velocity.  Then  the  velocity  at  any  point 
will  be  \^ 2g{x -\- h').  Hence,  in  this  case,  we  must  minimize 
the  expression 

Jx,  Vx^h'  ^  ^ 

Now  varying  y  and  z  as  before,  transforming  the  terms  ol  the 
first  order  until  they  assume  the  form 

SU=  A  -  L,-^fJ^'Mdydx+£ySzdx,  (2) 

and  then  equating  M  and  N  severally  to  zero,  we  have 


dx  V{x-\-h'){i+y+z") 

dx  |/(^:+iy(  I +/'  +  /') 


(3) 


BRACHISTOCHRONE   IN  SPACE.  373 


r  =  ^'-  (4) 


^{x  +  //)  (I  +/-+  z'^)  V{x  +  /.')  (I  +/"+  ^") 

*  Now  dividing  the  first  of  equations  (4)  by  the  second,  we  find 

that-^  must  be  constant.     This  is  sufficient  to  show  that  the 

curve  required  must  be  a  plane  curve,  and  hence  we  know 
that  the  solution  must  be  a  cycloid. 

Thus  we  see  that  we  can  sometimes  avoid  the  necessity  of 
integrating  completely  the  equations  M—o  and  iVr=o,  by 
showing  that  the  problem  can  be  reduced  to  one  of  two  co- 
ordinates ;  and  indeed  we  could  evidently  have  done  the  same 
thing  in  the  preceding  problem.  But  when  we  come  to  con- 
sider the  terms  of  the  second  order  we  must  evidently  resume 
three  co-ordinates,  because  we  now  require  that  the  primitive 
curve  shall  be  compared  wnth  all  curves  which  can  be  derived 
from  it  by  infinitesimal  changes  in  y,  z,  y' ,  z\  etc.,  some  of 
which  may  not  be  plane  curves,  and  would  be,  therefore,  ex- 
cluded by  the  employment  of  two  co-ordinates  only.  But  if 
we  compare' the  form  of  Fin  this  and  the  preceding  problem, 
observing  that  x  -\-  h'  has  no  variation,  and  denote  by  5  the 
coefficient  of  dx  under  the  integral  sign  in  equation  (11),  Art. 
307,  it  will  at  once  appear  that  these  terms  must  become 


2Jxa 


x-^     S  dx 

'^0      \/x  +  / 


which  must  be  also  essentially  positive,  since  \^x  -f-  h'  is  posi- 
tive throughout  U. 

312.  Now  since  we  know  that  the  required  cycloid  must 
have  the  line  joining  its  cusps  parallel  to  the  horizontal  plane 
xy,  and  itself  be  in  a  vertical  plane,  its  general  equation 
need  involve  but  five  constants.  For  we  have  to  consider 
the  three  co-ordinates  'of  one  of  the  cusps,  the  angle  which 


374  CALCULUS  OF    VARLATLONS. 

the  line  joining  these  cusps  makes  with  the  plane  of  xy,  and, 
lastly,  the  radius  of  the  generating  circle.  If  we  suppose  h 
zero,  and  the  cusps  to  lie  in  the  plane  of  yz,  these  constants 
will  be  reduced  to  four.  But  we  have,  as  the  most  general 
form  of  the  terms  at  the  limits, 

L,-L,^  V,dx,  -  V,dx,  +  h,dy^  -  h,dy,-\.H,dz,  -  IIJz,  =  o; 

so  that  it  appears,  as  before,  that  if  the  limiting  values  of  x  be 
fixed,  we  shall  have  just  the  requisite  number  of  conditions 
for  the  determination  of  the  constants  ;  and  that  if  these  limit- 
ing values  be  not  fixed,  we  must  restrict  dx^  and  dx,.  If  k'  be 
not  zero,  it  is  at  once  determined  by  the  initial  velocity  ;  but 
we  have  only  shown  that  the  cycloid  gives  a  minimum  when 
the  limiting  values  of  x  are  fixed. 

3(3.  Let  us  now  consider  briefly  how  many  constants  will 
occur  in  the  general  solution  of  problems  of  this  class,  and 
what  are  our  means  for  determining  them,  as  these  are  the 
only  points  which  need  any  additional  general  explanation. 

Assume  the  equation  U^=J^     V^^,  where  F is  any  function 

of  X,  y,  z,  y,  .  .  .  .  y^^",  z,  ....  ^^).  Then,  proceeding  in  the 
usual  way,  we  obtain  for  the  general  solution  the  two  differ- 
ential equations  M=  o  and  N  =  o.  Now  M  is  of  the  order 
2n  in  y,  and  ;/  +  m  in  z,  and  N  is  of  the  order  27u  in  z,  and 
m  -\-n  in  y.  Differentiating  N  2m  times,  and  M  n  -\-  in  times, 
we  shall  have,  together  with  M  and  iV,  3;;^  +  ^/  +  2  differential 
equations ;  the  highest  differentials  involved  in  any  of  these 
equations  bemg  2{in  -f-  11)  in  j/,  and  yri  -\-  nin  z.  Now  elimi- 
nating z  and  its  yn  +  71  differential  coefficients,  we  shall  obtain 
an  equation  in  x  and  y  only,  and  the  differential  coefficients  of 
y  with  respect  to  x.  The  order  of  this  equation  must  be 
2{m  -\-  n),  and  its  complete  integral  must  therefore  involve 
2{7n  +  n)  arbitrary  constants,  which  is  the  number  which  must 
be  contained  by  the  general  solution. 


PROBLEM  OF  LEAST  ACTION.  375 

Now  if  we  examine  the  most  general  form  of  the  limiting 
terms  L^  —  Z^,  it  will  at  once  appear  that,  unless  some  re- 
striction be  imposed,  there  must  be  as  many  independent 
terms  as  there  are  quantities  dx^,  dx^,  dy^,  dy^,  Sy/,  dy/,  .... 
<3>/^-i\  d^jo(^-i),  d^„  (^^0,  ....  (^^,(^-1),  (^ir/"^-i),  the  number 
of  which  will  be  2{m  -^  n)-\-2,  or  merely  2{m  -\-  n),  if  the  lim- 
iting values  of  x  be  fixed,  or  if  dx^^  and  dx^  be  restricted  as 
formerly.  Moreover,  it  will  appear,  as  before,  that  any  con- 
dition which  causes  one  of  these  equations  to  disappear  will 
itself  furnish  a  new  equation  of  condition,  so  that  the  number 
of  limiting  equations  will  still  remain  equal  to  that  of  the 
arbitrary  constants. 

Nevertheless  it  is  easy  to  see  that  the  reasoning  here  em- 
ployed may  be  subject  to  exceptions  similar  to  those  which 
have  been  explained  in  the  case  of  two  co-ordinates ;  but  these 
will  give  the  reader  no  serious  difficulty. 

3(4.  We  may  now  consider,  as  being  somewhat  connected 
with  our  subject,  the  principle  of  least,  or  more  properly  mini- 
mum action,  particular  cases  of  which  have  been  already  dis- 
cussed. 

Problem  L. 

A  particle  is  to  move  in  space  from  one  fixed  point  to  another, 
its  motion  being  controlled  solely  by  a  system  of  incessant  forces. 
Then  x,  y,  and  z  being  the  co-ordinates  of  any  point  of  its  path, 
ds  an  element  of  tins  path,  and  v  the  velocity  of  the  particle  at  the 
end  of  any  time  t,  it  is  required  to  show  that  the  7iature  of  this 
path  must  be  such  as  to  render  S  U  to  the  first  order  zero,  where 

Denoting  by  X,  Y  and  Z,  as  usual,  the  aggregated  com- 
ponents of  all  the  forces  in  the  direction  of  the  axes  of  ^,  j. 


37^  CALCULUS   OF  VARLATLONS. 

and  z  respectively,  we  shall  assume  the  well-known  equation 
in  mechanics, 

V dv  =  -  div")  =  Xdx  +  Ydy  +  Zdz,  (i) 

Now  if  we  suppose  the  particle  to  be  moving  along  the  re- 
quired path,  the  symbol  d,  as  applied  to  any  quantity,  denotes 
the  change  which  that  quantity  undergoes  when  the  indepen- 
dent variable,  which  we  may  here  assume  to  be  x,  receives  an 
infinitesimally  small  increment,  the  curve  remaining  unchanged. 
But  if  we  draw  any  derived  curve,  and  suppose  the  particle 
could  pass  from  any  point  /  on  the  primitive  to  some  point  P 
indefinitely  near/,  but  on  the  derived  curve,  then  if  we  give 
to  the  symbol  d  the  meaning  already  explained,  we  may  denote 
by  8  the  corresponding  change  which  the  various  quantities 
would  undergo  if  the  particle  could  pass  from/  to  P. 

Now  in  passing  from  /  to  P,  just  as  in  passing  along  any 
element  of  the  primitive  curve,  we  may  assume  that  X,  V, 
and  Z  remain  constant ;  and  hence,  denoting  by  ^^,  v^  and  v^ 
the  components  of  v  in  the  direction  of  x,  y  and  z  respectively, 
if  we  add  to  x,  y,  or  z  any  infinitesimal  increment,  the  corre- 

sponding  change  in  ^ — -,  etc.,  would  be  X,  Y,  or  Z  multiplied 

by  those  increments  respectively,  whether  those  increments 
were  added  as  differentials  for  the  purpose  of  enabhng  us  to 
pass  from  one  point  to  another  on  the  primitive  curve,  or  as 
variations  for  the  purpose  of  enabling  us  to  pass  from  any 
point  on  the  primitive  curve  to  an  adjacent  point  lying  on 
some  derived  curve. 

But  we  have  seen  that  any  derived  curve  can  be  obtained 
without  varying  x,  and  we  shall  therefore  consider/  and  Pas 
having  the  same  abscissa  x.  Hence,  accents  below  denoting 
differentiation  with  respect  to  /,  and  those  above  with  respect 
to  X,  and  remembering  that  Y:=y^^  and  Z  —  z^^,  (i)  may  be 
written 


PROBLEM   OF  LEAST  ACTLON.  377 

i  S  iv")  =v6v=  Y6y  +  Zdz  =  fjj  +  ^,/.5'.  (2) 

But  the  last  member  of  (2)  equals 

(x^7 + ^M^  -  ky.^y^  +  ^/^^.)-  (3) 

Now  we  have 

2       ds^       dx^    ,    dy    ,    dz^  21212  /  N 

Then  varying  i)^  under  the  supposition  that  neither  dt  nor  dx 
undergoes  any  change  from  variations,  we  shall  obtain 

vSv=y^dy^^z^dz^.  (5) 

That  this  supposition  may  be  made  will  appear  if  we  re- 
member that,  in  passing  from  p  to  P,  v  undergoes  no  change, 
so  that  dx  and  dt  for  that  element  of  the  curve  maintain  to 
each  other  whatever  ratio  they  had  before  the  curve  was 
varied.  Of  course  if  we  divide  the  whole  time  /  into  equal 
parts  dt,  the  corresponding  differentials  of  x  cannot  be  sup- 
posed to  be  equal  among  themselves ;  but  this  inequality  can 
in  no  way  affect  our  problem. 

Hence,  admitting  the  validity  of  (5),  equation  (3)  becomes 

{yfy  +  ^j^^)/  —  "^  <^^, 
and  (2)  may  therefore  be  written 

2vdv  =  d{7>')  =:  {ySy  +  z^6z)^.  (6) 

But  since  v  —  ---,  we  have  v^  =  ——.     Hence  (6)  gives,  after 
clearing  fractions, 

d{vds)  =  d{yfy-\-zSz).  (7) 

But  since  the  particle  is  to  pass  from  one  fixed  point  to 
another,  the   derived   curve    must   also   pass  through   these 


378  CALCULUS  OF  VARIATIONS,      ' 

points,  and  we  are  not  to  suppose  the  particle  capable  of  any 
displacement  at  either  point,  so  that  the  variations  of  y  and  z 
vanish  a1  both  these  points.  Moreover,  although  we  have 
really  regarded  t  as  the  independent  variable,  we  may  inte- 
grate (7)  as  though  that  variable  were  x.  For  d  in  (7)  denotes 
the  change  which  y^dy-{-zfiz  undergoes  in  the  time  dt,  or 
while  the  particle  passes  from  a  point  whose  abscissa  is  x  to 
one  whose  abscissa  is  x  ^  dx\  so  that  it  is  the  same  thing 
whether  we  suppose  these  changes  to  be  summed  up  for  the 
time  /j  —  t^  or  through  the  distance  x^  —  x^.  Therefore,  by 
integration,  (7)  gives 

=  {y,^y  +  ^/^).  -  {yfy  +  ^M-  =  o.      (s) 

316,  To  guard  against  certain  misconceptions,  we  observe, 
first,  that  the  reasoning  here  employed  would  not  be  applicable 
if  the  particle  were  compelled  by  a  system  of  forces  to  move 
along  a  fixed  material  curve.  For  then,  although  equation  (i) 
would  hold,  equation  (2)  would  not,  because  that  portion  of 
X,  Y  and  Z  which  arises  from  the  normal  pressure  of  the 
curve  upon  the  particle  would  vanish  for  any  point  P  without 
the  curve ;  so  that  we  could  not  say,  as  formerly,  that,  in  pass- 
ing from  p  to  P,  X,  V,  and  Z  would  remain  constant. 

We  observe,  secondly,  that  although  the  principle  just 
established  is  commonly  called  that  of  least  or  minimum 
action,  the  name  is  not  warranted,  at  least  by  the  preceding 
demonstration.  For  our  approximations  were  carried  to  the 
first  order  only ;  so  that  we  are  merely  able  to  say  that  the 
required  curve  must  be  such  as  to  render  (^f/to  the  first  order 
zero.  But  we  have  already  seen  that  the  terms  of  the  second 
order  in  (^^  do  not  always  become  positive,  but  sometimes 
vanish  also,  in  which  cases  we  inferred,  although  we  did  not 
investigate  the  matter,  that  .those  of  the  third  order  would  not 
likewise  vanish,  and  that  therefore  d^  might  have  either  sign 


SHORTEST  LINE   ON  A    SPHERE.  379 

at  our  pleasure,  thus  showing  that  U  could  be  neither  a  maxi- 
mum nor  a  minimum.  It  will  be  found,  however,  that  the 
terms  of  the  second  order  vn  dU  never  become  negative,  and 
indeed  it  is  generally  conceded  that  the  action  can  never,  as 
Lagrange  erroneously  supposed,  become  a  maximum. 


Section  II. 

case  in  which  the  variations  are  connected  by 
equations,  differential  or  other. 

Problem  LI. 

316.  It  is  required  to  determine  the  nature  of  the  lifie  of  min- 
mum  length  which  can  be  draivn  between  two  fixed  points  or  curves 
on  the  surface  of  a  sphere. 

Here 

U =.C"  VT+7'"=-+7^ dx  =/;■  Vdx ; 

and  taking  the  variation  of   U,  and  integrating  in  the  usual 
manner,  we  have 


'^^»      dx  Vi+/'  +  z"  ^'^      dx  Vi-\-y"  +  z" 

=  hfiy,  -  hfy,  +  HM  -  HM 

+  r^MSydx  +  f'NSzdx  =  o.  (i) 


380  CALCULUS   OF  VARIATLONS. 

Now  in  this  case  the  variations  of  y  and  z  are  not  indepen- 
dent, all  derived  curves  which  cannot  be  drawn  upon  the  sur- 
face of  the  given  sphere  being  excluded  from  comparison  with 
the  primitive.  Nevertheless  it  is  evident  that  the  integrated 
and  the  unintegrated  parts  of  (^^must  severally  vanish.  For 
we  may  suppose  each  part  to  be  expressed  in  terms  contain- 
ing one  variation  only,  the  other  having  been  eliminated. 
Hence  we  have 

L,-L,  =  o,         r\Mdy-\-N6z)dx^o,  (2) 

But  we  have  from  the  sphere 

x''+/-^z'  =  r\         ydy^zdz  =  0,         d^  =  IlZ^»    (3) 
Hence  (2)  may  now  be  written 

X"  1  ^-  ?  1  '^"^^  =iy''y<^^ = °-        (4) 

Whence  it  will  at  once  appear  that  to  maximize  or  minimize 
U,  M '  must  vanish.     Equating  —  M'  to  zero,  we  have 

d  y'  d  z'  y 


dx  Vi+y  +  z''      dx   Vi+y  +  z"   z 
so  that  we  obtain 

1  d  ---J-.=  =  i  d-—J--= .  (5) 

317.  Before  proceeding,  we  shall  find  it  necessary  to 
change  the  mdependent  variable  to  s.  It  is  evident  that  (5) 
may  be  written 

I    ,dy  '    I    ,dz  ,^. 

-d-^-  =z~d  --.  (6) 

y     ds       z     ds 


SHORTEST  LINE   ON  A    SPHERE.  38 1 

Although  the  symbol  d  in  (6)  denotes  change  incident  upon 
changes  in  x,  yet  we  were  not  originally  bound  to  consider 
two  consecutive  values  of  dx  as  absolutely  equal,  and  we  may 
therefore  suppose  that  these  differentials  were  so  taken  as  to 
make  those  of  s  always  equal.     Hence,  regarding  ds  as  always 

constant,  multiplying  by  --,  and  denoting  by  accents  differen- 
tiation with  respect  to  ^,  (6)  becomes 

y 

Now  multiply  both  the  numerator  and  denominator  of  the 

first  fraction  by  y' ,  and  of  the  second  by  z' ,  and  denote  the 

A  C 

resulting  fractions  by  —  and  — ,  which  are,  of  course,  equal  to 

each  other  and  to  the  members  of  (7).  Hence  the  quantities 
A,  By  C  and  D  are  in  proportion,  and  therefore 

A  +  C:B  +  D::A  :  B::C'.  Dwy" ly-z^' iz. 

Hence  either  member  of  (7)  equals 


B-\-D        yy+  zz' 


(8) 


But  from  the  equation  of  the  sphere,  and  also  the  equation 

x'"" -{- y -^  z''' —  I,  we  have 

xx'  +  yy'  +  zz'  =  o,        x'x"  rf-  y'y"  +  z'z"  =  o : 


and  therefore  (8)  becomes  — ,  so  that  we  have 

X 


X"       y"       z" 

^  =  -J  =  -z'  (9) 


382  CALCULUS   OF  VARIATIONS. 

Now  because  (9)  is  true  we  may  evidently  write  the  following 
three  equations : 

xd"^)!  —  yd^x  =  o,         xd^^  —  ^d^x  =^  o,         j/d^j2  —  zd'^y  =  o. 

Integrating  these  equations  by  parts,  we  obtain 

xdy  —  ydx  =  a' ^         xdz  —  zdx  =  b' ,        ydz  —  zdy  =  c' .    (10) 

Multiplying  the  first  of  these  equations  by  z,  the  second  by 
—  r,  the  third  by  x,  and  adding  the  products,  we  shall  obtain 

a'z  —  b'y  +  c'x  —  o.  (11) 

In  this  equation  the  constants  are  infinitesimal,  but  dividing 
by  one  of  them,  as  c' ,  the  two  resulting  constants  may  have  any 
value  we  please,  infinitesimal,  finite,  or  infinite ;  and  we  may 
write  X  -\-  ay  -\- bz  ^  o,  the  equation  of  a  plane  passing  through 
the  centre  of  the  sphere.  The  required  curve  must,  therefore, 
be  a  great  circle. 

3t8.  To  determine  the  constants  a  and  b,  we  have,  if  we 
suppose  the  curve  to  connect  two  fixed  points,  the  equations 

-^i  +  ^7i  +  ^-s",  3=  o,         x,-\-ay,-^bz,^o. 

But  suppose  the  curve  is  to  connect  two  given  curves,  and 
let  the  equations  of  the  curve  for  the  upper  limit  be 

y=f{x)=f,  z=zF{x)  =  F,  or  dy=zfdx,  dz^F'dx.  ^12) 
Then  we  shall  have,  as  in  Art.  69, 

^y.  =  {f'  ~-y'\dx,        and     dz,={F'  -z'\dx,,         (13) 

Also  it  is  evident,  as  before,  that  L,  and  L,  must  severally 
vanish,  and 


L,=   V{i-^y''-\-z'\dx, 


SHORTEST  LINE   ON  A    SPHERE.  383 

Eliminating  8y^  and  dz^  by  means  of  (13),  and  reducing,  we 
obtain  i  -{- y^  fl  +  z^  F^  =  o,  which  shows  that  the  great  cir- 
cle must  cut  the  limiting  curve  at  right  angles ;  and  a  similar 
result  can  evidently  be  obtained  for  any  curve  at  the  lower 
limit. 

If  we  suppose  the  limiting  values  of  x  only  to  be  fixed,  we 
shall  obtain  for  either  limit,  after  having  eliminated  dz  by  (3), 
zdy  ~  ydz  =  o  =  —  c' ,     Hence  (li)  becomes 

a'z  —  b'y  =  o  =  z  —  a"y,         or     z  —  a"y, 

where  a"  remains  undetermined,  as  it  should,  it  .being  the  tan- 
gent of  the  inclination  of  the  great  circle  to  the  plane  of  xy. 
We  conclude,  therefore,  that  the  great  circle  must  be  so  drawn 
that  its  intersection  with  the  plane  of  xy  shall  always  coin- 
cide with  the  axis  of  x, 

319.  It  will  appear  by  reference  that  U  has  here  the  same 
general  form  as  in  the  first  problem  of  this  chapter ;  and  hence 
if  the  limiting  values  of  x  be  fixed,  the  terms  of  the  second 
order,  as  they  at  first  arise,  will  be  the  same  as  in  equation 

(11),  Art.   307,  which  may  be  written  dU—-J     Sdx.    But 

now  the  mode  of  eliminating  Sz  must  be  rendered  exact  to  the 
second  order,  and  for  this  purpose  we  have 

6z  =  ^Uy  +  l^S/=-y-6y-l+^Sy\ 
dy  -^^  2df    ^  z    ^  2z'       ^' 

which  will  at  once  appear  if  we  remember  that  x  has  no  vari- 
ation, and  that  Sy  and  Sz  are  taken  along  any  section  of  the 
sphere  at  right  angles  to  the  axis  of  x.  Substituting  this  value 
of  Sz,  we  shall  evidently  obtain,  as  the  coefficient  of  Sy,  M' ,  as 
before,  which  must  be  equated  to  zero  as  formerly.  But  since 
we  must  not  reject  the  new  term  of  the  second  order  arising 


3^4  CALCULUS  OF  VARLATIONS. 

from  the  elimination  of  dz,  we  add  it  to  those  already  in  the 
second,  and  the  complete  terms  then  become 

When  the  limiting  values  of  x,  y  and  z  are  fixed,  and  the  arc 
joining  the  two  fixed  points  is  less  than  a  semi-circumference, 
the  sign  of  these  terms  is  undoubtedly  positive,  as  we  know 
from  other  considerations  that  we  have  a  minimum.  Never- 
theless the  author  is  unable  to  present  any  satisfactory  general 
demonstration  of  the  fact  that  these  terms  fulfil  all  the  neces- 
sary  conditions  for  a  minimum. 

320.  The  method  employed  in  the  preceding  problem  is 
not  sufficiently  general  for  all  cases,  since  it  is  evident  that  the 
connecting  equation  may,  as  when  s  is  the  independent  vari- 
able, be  an  unintegrable  differential  equation,  which  will  not 
enable  us  to  express  dz  in  terms  involving  dy  only.  In  this  case 
we  must  adopt  the  method  of  Lagrange,  with  which  the 
reader  is  already  partially  familiar,  and  which  we  will  now 
briefly  explain  in  a  somewhat  more  general  manner. 

321.  Suppose  we  seek  to  maximize  or  minimize  the  expres- 
sion U  =    I ^  Vdx,  where  V  is  any  function  of  x,  y,  y\  .  .  .  . 

z,  z\  .  .  .  .;  and  suppose  also  that  the  equation  f{x,  y,  z,  /,  z' , 
....)  —  o—  f  \s  always  to  hold.  Then,  because/  is  always 
zero,  df  must  vanish  ;  that  is,  we  must  have 

fy^y  +  fy'^y  +  etc.  +f,6z  -\-A>Sz'  +  etc.  =  o.  (i) 

Then  /  being  any  quantity,  constant  or  variable,  we  may  write 
/     Idfdx  =  o.     Now  vary  U  to  the  first  order,  equate  SU  to 

t/  Xq 

zero,  and  transform  by  integration  as  usual.     Then,  in  like 


METHOD   OF  LAGRANGE.  385 

manner,  transform  /    Udfdx,  and  add  the  result  to  dU.    Then 

giving  only  the  general  form  of  the  terms  free  from  the  inte- 
gral sign,  we  shall  have 

6U  =  \Vy,  +  lfy,  -  ( F,. +  //,.)'+  etc.  KJ.  + etc. 
+{Vz'+  Ifz'  -  ( V,..  +  If,.?)'  +  etc. \ Sz  +  etc. 
^-rWv^-lfy  -  (F^ +//,-)'  + etc.]  tfj 

+  Wz^  Ifz  -  {Vz'+^/z'Y  +  etc.]  ^^}d;t:  =  o,      (2) 

Now  whatever  be  the  value  of  /,  the  integrated  and  the 
unintegrated  parts  oi  ^U  must  severally  vanish.  Then  if  we 
assume  /  so  as  to  make  the  coefficients  of  either  of  the  quan- 
tities ^j/dx  or  S^dx  vanish,  the  coefficient  of  the  other  must 
vanish  also.  Thus  we  reduce  (5^^  to  such  a  form  that,  without 
eliminating  either  Sj/  or  ^^,  we  may,  in  the  unintegrated  part, 
regard  these  quantities  as  if  they  were  really  independent,  and 
equate  their  coefficients  severally  to  zero.  But  it  is  evident 
that  before  we  can  obtain  j/  and  ^  as  functions  of  x,  we  must 
be  able  either  to  eliminate  /  or  to  determine  it  also  as  a  func- 
tion of  X,  y  and  s.  This,  however,  can  in  general  be  accom- 
plished, because,  in  addition  to  the  differential  equations  ob- 
tained by  equating  to  zero  the  coefficients  of  6y  and  Sz,  we 
now  have  the  equation /=  o. 

322.  The  arbitrary  constants  which  enter  the  general  solu- 
tion must  evidently  be  determined  by  the  conditions  which 
are  to  hold  at  the  limits.  Denoting  the  terms  at  the  limits  by 
Zj  —  Zq,  we  may  evidently  in  general  equate  these  quantities 
severally  to  zero.  Then  in  Z,,  for  example,  the  value  which 
we  have  been  obliged  to  assign  to  /  will  not  usually  cause  the 
coefficients  of  dx\,  Sy^,  6y^\  Sz^,  Sz^\  etc.,  to  vanish  severally. 
Moreover,  these  variations  are  not  independent,  because  the 


386  CALCULUS  OF  VARLATIONS. 

equation /=  o  is  to  hold  among  them;  so  that  we  cannot 
equate  these  coefficients  severally  to  zero. 

We  see,  therefore,  that  although  we  shall  have  as  many 
equations  at  the  limits  as  there  are  independent  variations,  a 
general  discussion  of  the  number  of  the  arbitrary  constants 
involved  in  the  general  solution,  and  of  the  number  of  the 
ancillary  equations  for  their  determination,  must  become  com- 
plicated. Some  results  relative  to  these  points  have  been 
obtained  by  Prof.  Jellett,  and  are  given  in  Art.  59  of  his  work, 
which  results  appear  to  be  correct,  although,  as  he  himself 
states,  they  are  at  variance  with  some  obtained  by  Poisson. 
It  will  here  be  sufficient  to  present  Prof.  Jellett's  conclusions 
without  demonstration. 

Let  V  be  of  the  order  n  in  y  and  m  in  ^,  and  let  the  con- 
necting equation  /  =  o  be  of  the  order  n'  in  y  and  m'  in  z,  the 
limiting  values  of  x  only  being  fixed. 

Then,  first  supposing  n  ^  n'  and  in  >  ;;/,  the  number  of 
constants  involved  in  the  general  solution  will  be  the  greater 
of  the  two  quantities  2{m  -\-  n')  and  2(;//  -|-  71),  while  the  num- 
ber of  the  independent  variations  remaining  in  L^  —  L^,  whose 
coefficients  may  be  equated  to  zero,  will  be  the  same  ;  so  that 
all  the  constants  can  in  this  case  be  determined ;  and  the  same 
conclusion  holds  when  in  >  in'  and  n  <  n'. 

If  we  next  suppose  n  <  it'  and  in  <  in',  the  number  of 
constants  involved  in  the  general  solution  will  in  general  be 
2{m'  -\-  n'),  and  of  these  constants  there  may  remain  undeter- 
mined any  number  not  exceeding  the  lesser  of  the  two  quan- 
tities 2{in'  —  in)  and  2{n'  —  n). 

323.  The  method  of  Lagrange  is  capable  of  extension  to 
any  number  of  dependent  variables.  For  example,,  let  V  con- 
tain jr,  y,  z,  u,  and  any  differential  coefficients  of  y,  z  and  u 
with  respect  to  x ;  and  let  the  equations 

fix,  y,  z,  u,  y',  z',  ii' ,  etc.)  =  o=f 
and 

F{x,  y,  z,  u,  y,  z',  ic',  etc.)  =0  =  F 


METHOD   OF  LAGRANGE.  3^7 

always  hold.  Then  both  d/and  (^/^must  vanish;  and  assum- 
ing /,  as  another  undetermined  quantity,  we  have 

r^lS/dx  =  o       and      r^LdEdx  =  o. 

Adding-  both  these  equations  to  dV,  and  transforming  as  usual, 
the  integrated  and  the  unintegrated  parts  must  severally  van- 
ish.    Then  we  may  write 

r\MSy  4-  N6:2  +  Pdu)  dx  =  0\ 

and  if  we  so  assume  /  and  /^  as  to  cause  any  two  of  the  quan- 
tities M,  N  and  P  to  vanish,  the  other  must  vanish  also,  after 
which  /  and  /^  must  as  before  be  eliminated,  or  found  as  a 
function  of  x,  y,  z  and  ?/,  in  order  that  we  may  obtain  a  com- 
plete solution. 

324.  If  we  adopt  the  method  of  Lagrange  in  the  preced- 
ing problem,  we  shall  obtain  the  equations 


ly =r  =  o,         Iz -  =  o, 


'      y'      -0, 

Iz- 

d              z' 

dx  Vi+y+z'' 

dx  Vi+y+z'' 

so  that  eliminating  /  we  shall  arrive  at  equation  (6),  Art.  3 1 7. 
Moreover,  /  will  not,  in  this  case,  appear  at  all  in  the  terms  at 
the  limits,  which  will  therefore  be  of  the  same  form  as  before. 

325.  The  last  example  is  merely  an  individual  of  an  exten- 
sive class,  and  was  discussed  separately  merely  for  the  sake  of 
introducing  in  a  simple  manner  the  subject  of  the  present 
section.  We  shall  now  proceed  to  consider  a  very  general 
problem  given  by  Prof.  Jellett,  from  the  partial  solution  of 
which  the  last  and  many  other  examples  can  be  readily  solved. 


3^^  CALCULUS  OF  VARIATIONS, 


Problem  LII. 


Let  V  be  any  function  of  x,  y  and  z,  and  let  these  quantities  be 
also  conjiected  by  the  equation  f{x,  y,  z)^^  o  =:  f  Then  it  is  re- 
quired to  maximize  or  7iiininiize  the  expression 


U^  r\Vi-\-y"  +  z''dx, 


Supposing  y  and  z  found  as  functions  of  x,  we  may  evi- 
dently then  regard  x,  y  and  z  as  the  co-ordinates  of  some 
curve ;  so  that  we  may  consider  as  usual  that  we  require  a 
curve  whose  co-ordinates  shall  be  the  values  oi  x,  y  and  z  in 
the  general  solution,  and  which  we  may  therefore  call  the  re- 
quired curve.  Moreover,  since  the  equation /=o  may  be 
regarded  as  the  equation  of  a  surface,  we  may  suppose  that 
the  curve  is  required  to  lie  upon  this  given  surface. 

Now  let  ds  be  an  element  of  this  required  curve.  Then 
we  may,  as  in  the  case  of  two  co-ordinates,  adopt  s  as  the  in- 
dependent variable,  considering  x,  y  and  z  as  functions  of  s, 
which  itself  receives  no  variation.  We  must,  however,  in 
this  case,  adopt  the  method  of  Lagrange  for  three  dependent 
variables.  For,  since  s  is  to  be  the  independent  variable,  x,  y 
and  z,  besides  satisfying  the  explicit  equation  /=  o,  must  also 
satisfy  the  unintegrable  differential  equation 


dx""    .    dy''       dz^  „ 


326.  Now  transforming  to  s,  we  have 


CURVES  ON  SURFACES. 


389 


(2) 


dU  =  Z\  ds^  —  V^ ds^  -[-    /      (^a;^-^  +  "^y^y  +  ^2 ^^)  ds  —  O, 

j(''V  {x'dx'  +  ^'^j/'  +  ^'(^^0  ds  =  o, 

I/Sq 


where  accents  above  denote  differentiation  with  respect  to  s, 
and  hteral  sufhxes  partial  differentiation  as  hitherto.  Then, 
proceeding  as  usual,  we  obtain 


^^x  +  IJx  -  {Ix')'  =  O  -  7;,  +  I  J,  -  Ix"-  X'l\ 

""^y  +  ^Jy  —  Wy  ^o  =  Vy-\-  IJy  -  ly"  -  y'l',  >■ 

'Vz + /./.  -  (i^'y  =  0  =  2^.  +  Kfz  - 1^"  -  ^'i'- . 


(3) 


Multiplying  these  equations  by  x',  y'  and  z'  respectively,  and 
adding  the  products  employing  the  equations 

fxx'-\-fyy'^f,z'^f  =  o,        x'^+y'^  +  z'^=ir 

v^  x'  +  Vyy'  +  v^  z'  =  v\       x'x"  +  y'y'''  +  z'z"  =o\     (4) 

lxx'-\-lyy  -^hz'  =l\  '     ^ 

we  have 

v'  —  I'  =  0        and     l=v-{-a,  (5) 

327.  Before  proceeding,  we  must  determine  the  constant 
a,  and  this  will  lead  us  to  examine  the  terms  at  the  hmits. 
These  terms  are 

v^  ds,  —  V,  ds,  -\-  l,{x'dx  -\-ySy  +  z'Sz\ 

-  IXx'dx  +y'Sy  +  z'S^\  =  o.         (6) 

Now  if  the  required  curve  is  to  connect  two  fixed  points, 


390  CALCULUS   OF  VARIATIONS, 

we  shall  have,  by  reasoning  like  that  employed  in  Arts.  276 
and  2"]"],  for  either  limit, 

Sx  =  —  x'ds^         (Sj/  =  —  yds,         and     d^  =:  —  z'ds,       (7) 

Kence,  substituting  these  values  in  (6),  and  observing  the 
second  equation  (i),  we  have 

{y —  l)^ds^  —  {y  — l),ds^  —  0, 
so  that 

v^=^  L      *    and     I  =  v. 

But  if  the  required  curve  is  to  connect  two  fixed  curves, 
let  the  equations  for  the  fixed  curve  at  the  upper  limit  be 
dy-=.pdx  and  dz=^qdx.  Then,  by  reasoning  precisely  like 
that  of  Art.  278,  we  have 

^y.  +y^ds,  =  pSpx^  +  x^ds^,  ) 

\  (8) 

<^^j  4-  z^ds^  =  ^1(^-^1  +  x^'ds^.  ) 

Substituting  in  L^  the  values  of  Sy^  and  dz^  found  from  (8),  we 
have,  omitting  sufhxes, 

V  ds  -{-  Ix'dx  -j-  ly\p  Sx  -\-  px'ds  —  y'ds) 

-\-  lz\q  6x  -\-  qx'ds  —  z'ds)  =  o.  (9) 

But  ds^  and  dx^  are  entirely  independent,  so  that,  equating 
their  coefficients  severally  to  zero,  we  have 

V  +  ly'x'p  —  ly  +  Iz'x'q  —  Iz""  =  o,  ) 

(10) 
lx'^ly'p^lz'q=o.) 

Multiplying  the  second  of  these  equations  by^;ir',  and  subtract- 
ing from  the  first,  observing  the  second  of  equations  (i),  we 
find,  as  before,  that  /,  —  %\  and  /—  v. 


CURVES  ON  SURFACES.  391 

Now,  supposing  /^  or  v^  not  to  vanish,  divide  the  second  of 
equations  (lo)  by  /j^/,  and  we  shall  obtain 

which  shows  that  the  required  curve  must  cut  the  fixed  curve 
at  right  angles  ;  and  a  similar  result  can  evidently  be  obtained 
for  another  fixed  curve  at  the  lower  limit. 

328.  Let  us  now  return  to  the  general  solution. 
Putting  V  for  /  in  (3),  we  have 


'^x  +  IJx  —  'VX"  —  X'v'=^  O,  ^ 

'^y  +  ^Jy  —  "^y"  —  7V=  o, 
Vz  +  IJz  —  'vz"  —  z'v'  —  o. 


(II) 


Let  A,  B  and  C  be  the  angles  made  with  the  co-ordinate 
planes  by  the  plane  of  that  normal  section  which  contains  at 
any  point  the  tangent  to  the  required  curve.  Then,  because 
the  plane  contains  the  normal  to  the  surface,  and  also  the  tan- 
gent to  the  required  curve,  we  must  have  the  equations 

fx  cos  A  -\-fy  cos  B  -\-fz  cos  C  =  oA 

\  (12) 

x^  cos  A  -{-  y  cos  B  -\-  z'  cos  C  —  o.) 

Hence,  multiplying  the  first  of  equations  (11)  by  cos^,  the 
second  by  cos  B,  the  third  by  cos  C,  and  adding  the  products, 
we  have 

Vx  COS  A  -j-  I'y  COS  B  -\-Vz  COS  C 

—  V  ix"  COS  A  ^  y"  COS  B  -{-  2"  COS  C)  =  o.  (13) 

Now  let  A^,  B^  and  C^  be  the  angles  which  r^,  the  radius  ot 
curvature  ot  the  required  curve,  makes  with  the  co-ordinate 
axes.     Then  it  is  known  that 


392  CALCULUS  OF  VARIATIONS. 

COS  A^  —  —  r^x" ,  cosB^  =  —  r^y\  cos  C,  =  —  r^z",  (14) 
Equation  (13)  may  therefore  be  written 

Vx  COS  A  -|-  %>y  cos  B-\-Vz  cos  C 

= (cos  A  cos  A^  -\-  cos  B  cos  B^  -\-  cos  6"  cos  C^).      (i  5) 

^/ 

Next  let  0  be  the  angle  which  the  osculating  plane  to  the 
curve  makes  with  the  plane  of  the  aforesaid  normal  sections. 
Then  it  is  known  that 

cos  A  cos  A  ^  -\-  cos  B  cos  B^-\-  cos  C  cos  C^  =  sin  0.       (16) 

Also,  r^^  being  the  radius  of  curvature  of  this  normal  section, 
we  have,  by  Meunier's  Theorem, 


sm'  0=1 '—. 


(17) 


Equation  (15)  may  therefore  be  written  under  any  one  of  these 
three  forms : 


-^ ,  —  -„-  (^a;  COSy4  +  Vy  COS  B  +  ^^COS  C)'\ 

r,        r,r      v 


sm  0 
r. 


{Vsc  COS  A  -{-  Vy  COS  B  -\-  VzCOS  C), 


tan  0            I  .  .    ,  „   ,  ^. 
= [Vx  COS  A  -{-  Vy  cos  B  +  "^'^cos  6 ). 


[      (18) 


The  first  two  equations  are  evident  enough,  but  to  obtam  the 
third  we  have 


r;  „  sm  o 

-L^=zCOS'o=- —' 

r,/  tan  o 


CURVES  ON  SURFACES.  393 

Hence 

or 

Whence 


^;: 

rj : :  sin''  o  :  tan''  o^ 

sin' 

'o:r; 

::tanV:r,;. 

sin''^ 

tan"^ 

r: 

r,; 

329.  Equations  (i8)  are  as  far  as  we  can  carry  the  general 
solution,  so  long  as  the  form  of  v  is  altogether  undetermined  ; 
but  we  may  nevertheless  deduce  a  remarkable  property  be- 
longing to  this  class  of  curves. 

Suppose  we  consider  the  integral  U^=J  —  =  /  v^ds. 
Then,  since 


dv^ 

I   dv 

dv,_ 

I  dv 

dv,_ 

I  dv 

dx 

v'  dx' 

dy 

v'dy' 

dz 

v'dz' 

the  equation  in  v^,  which  will  replace  (15),  will  reduce  to  equa- 
tion (15)  in  V,  except  that  the  sign  of  the  second  member  will 
be  changed.  Therefore  the  first  of  equations  (18)  includes  the 
solution  of  both  problems.  Hence,  because  equation  (15)  in 
V  can  be  converted  into  equation  (15)  m  v^  by  merely  chang- 
ing the  sign  of  r^,  we  conclude  that  if,  upon  a  given  surface,  a 
curve  whose  equation  is 

possess  the  property  of  rendenng  /  v  ds  2i  maximum  or  a 
minimum,  the  curve  whose  equation  is 

r,—  —f{x,y,  z,y:c) 

—  a  maximum  or  a 
minimum. 


394  CALCULUS  OF  VARIATIONS. 

330.  We  may,  before  considering-  particular  cases,  deduce 
another  property  of  this  class  of  curves. 

Let  d  be  a  fixed  point  and  TT'  a  fixed  curve,  both  being 
situated  upon  a  given  surface,  and  let  the  arc  TT'  be  taken 
indefinitely  small;  then  draw  two  curves  OT smd  0T\  each 
having  the  property  of  maximizing  or  minimizing  the  expres- 
sion U—  I  vds,  when  the  limits  are  fixed.     Then,  since  each 

curve  renders  U  a  maximum  or  a  minimum,  they  must  both 
satisfy  the  same  differential  equations ;  and  unless  we  suppose 
that  there  could  be  two  solutions,  (9rand  OT'  must  be  indefi- 
nitely near  at  each  point,  so  that  OT'  could  be  obtained  by 
varying  OT,  and  also  the  upper  limiting  value  of  x.  Therefore 
OT'—  (9 r  must  equal  that  portion  only  of  (^t/ which  is  with- 
out the  integral  sign — that  is,  since  the  lower  limit  remains 
fixed,  and  /^  =  v^ — must  equal 

v^  ds  +  v^  x/dx,  -4-  V,  yl^y^  +  z\  z^'Sz^.  (19) 

Denote  TT'  by  dS.     Then  we  must  have 

dz,  dz 

dz,  =  -77T-  dS,  —  —r  ds,. 
'       dS^  ds^     ^ 

Substituting  these  values  in  (19),  we  have 

OT'  -OT=v,-v,  {x"-\-y"  +  z'\  ds. 


CURVES  ON  SURFACES,  395 

But  denoting  by  t  the  angle  OT'T,  (20)  gives 

OT- OT  =  v,  cos  tdS,  (21) 

331.  From  the  second  of  equations  (18)  may  be  deduced 
two  others,  which  will  sometimes  be  found  useful. 

Let  A  J  J,  B^^  and  C^^  be  the  angles  made  by  the  osculating 
plane  with  the  co-ordinate  planes.     Then  we  have 

cos  A^^  =  r,  {z'y"  -  y'z"),         cos  B^^  =  r^  {x'z"  -  ^'x"\  ) 

[  (22) 
cos  C,,  =  r,  {y'x"  -  x'y).  ) 

Now  let  the  equation  of  the  given  surface  be 

ds  =  Zxdx  -\-  Zy  dy.  (23) 

Then  we  have 

Substituting  this  value  in  the  second  of  equations  (18),  and 
then  eliminating  either  x"  or  y"  by  the  equation 

x'x''-^yy'^z'z"=o, 

we  have  either  of  the  following  forms : 

x"-{-z^z"  = 


^    ""^  ^  {v:c  cos  A-\-VyCOS>B-\-  2;;jCOs  C)y, 


y"-^ZyZ-  = 


(25) 


-^--tl^:^  {v^  cos  A  +  Vy  cos  B  +  v^,  cos  C)x' .  ■ 


39^  CALCULUS  OF   VARIATIONS. 

332.  We  may  now  proceed  to  consider  some  particular 
cases  of  the  foregoing  theory. 

Problem  LIII. 

It  is  required  to  find  the  line  of  minimum  length  which  can  be 
drawn  upon  a  given  surface  between  two  fixed  points  or  two  fixed 
curves  situated  upon  the  same  surface. 

This  problem  is  the  simplest  case  of  the  preceding,  to 
which  it  may  be  reduced  by  writing  v  =^  i,  v^  =^  o,  Vy  ^=^  o  and 
Vz  =  o.  Whence  the  second  members  of  equations  (i8)  must 
vanish,  and  the  first  of  those  equations  will  give  r^  =  r^^,  which 
makes  o  vanish. 

We  see,  therefore,  that  curves  of  this  class,  or,  as  they  are 
generally  termed,  geodesic  or  geodetic  curves,  must  be  such 
that  their  osculating  plane  at  any  point  shall  be  perpendicular 
to  the  given  surface  at  that  point. 

Now,  in  Prob.  LI.,  since  the  radius  of  every  normal  sec- 
tion of  a  sphere  is  the  radius  of  the  sphere  itself,  it  at  once 
appears  that  every  geodesic  curve  drawn  on  its  surface  must 
be  the  arc  of  a  great  circle. 

We  shall  not,  however,  enter  upon  an  extended  discussion 
of  geodesic  lines,  but  shall  give  the  chief  points  concerning 
them,  following,  as  we  have  done  since  the  beginning  of  Prob. 
LIL,  the  guidance  mainly  of  Prof.  Jellett. 

333.  The  equation  of  a  geodesic  line  is  deducible  in  sev- 
eral ways. 

1st.  It  is  evident  that  all  the  reasoning  in  Prob.  LI.  would 
hold  if  the  equation  of  the  given  surface  were  f  —  o,  except 
that  X,  y  and  z  in  the  denominators  of  the  equations  would  be 
replaced  hy  fx,  fj  and  fz  respectively,  the  equations  remain- 
ing otherwise  unaltered.  This  would  hold  as  far  as  equation 
(9),  which  would  become 

X     y     z 

fx  fy         fz 


GEODESIC  CURVES.  397 

Hence  we  may  evidently  write  the  equations 

fx  dy  —  fyd'^x  =  o,        /p  d'^  —  f^d'^x  =  o,  ) 
fyd'^  - /,dy  =  O.  ) 

2nd.  If  the  equation  of  the  given  surface  be 
d^  =1  Zxdx  -\-  Zy  dy, 
then  equations  (25)  give 

x"  +  z^.  z"  =  o,        /'  +  Zy  z"  =  o.  (2) 

3rd.  Or,  because  <?  =  o,  we  have,  from  (24), 

z,  {/z'^-  z'y")  +  Zy  {z'x"  -  x'z")  =  x'y"  -y'x",  (3) 

But  we  have  the  equations 

dz  =  Zx  dx  -\-  Zy  dy, 
d^'z  =  Zxx  dx"  4-  2Zxy  dxdy-\-  Zyy  dy""  +  Sydy. 
Eliminating  dz  and  d^z  in  (3)  by  the  values  just  given,  it  becomes 

(I  ~T"  ^x   ~|    ^y  )  yxx  ~r  \^y       ^xyx)  \^xx    \    '^^xyyx  ~r  ^yyyxx)  =^  0> 

which,  together  with  the  equation  dz  —  Zxdx  —  Zydy  =z  o,  rep- 
resent the  geodesic  line. 

334.  As  another  property  connected  with  these  lines,  we 
may  notice  that  equation  (21),  Art.  330,  will  now  become 
OT^  —  OT—  cos  t  dS.  Now  it  is  easily  shown  b}^  the  differ- 
ential calculus  that  if  the  lines  OT  and  OT'  were  right  lines, 
the  point  O  and  the  curve  TT'  being  in  free  space,  the  above 
formula  would  hold.     Therefore  we  infer  that,  so  far  as  their 


39^  CALCULUS  OF    VARIATIONS. 

change  of  length  is  concerned,  we  may,  in  the  application  of 
infinitesimals,  regard  all  geodesic  lines  as  right  lines. 

But  we  must  here  note  a  difference  between  the  right  line 
and  most  other  geodesic  lines.  For  while  the  former  is  always 
the  line  of  minimum  length  between  two  fixed  points,  the  lat- 
ter are  not  in  every  case.  For  on  a  given  surface  suppose 
two  indefinitely  near  geodesic  lines  to  be  drawn :  these  lines 
will  in  general  intersect  at  some  point.  Here,  then,  we  would 
haye  between  two  points  two  indefinitely  near  geodesic  curves 
satisfying  the  same  differential  equations,  so  that  we  would 
naturally  infer,  as  in  the  case  of  two  co-ordinates  only,  that 
the  geodesic  line  ceases  to  be  a  minimum  when  the  integral 
U  ranges  from  one  intersection  to  the  other.  This  remark, 
although  undoubtedly  true,  can  only  be  called  an  inference, 
as  we  cannot  apply  Jacobi's  Theorem  with  any  success  to  this 
class  of  problems ;  still  the  principle  in  question  is  well  illus- 
trated in  the  case  of  the  sphere,  where  the  two  geodesic  curves 
will  be  two  indefinitely  near  semi-circumferences,  having  of 
course  the  same  length. 

Again,  we  have  already  seen  that  the  curve  which  renders 

vds  di  maximum  or  a  minimum  must,  if  one  or  both  ex- 


tremities  are  to  be  confined  to  fixed  curves,  cut  its  limiting 
curve  at  right  angles,  and  hence  this  must  be  true  also  of  geo- 
desic curves.  If,  therefore,  from  a  fixed  point  upon  a  given 
surface  geodesic  curves  of  a  given  length  be  drawn  in  every 
direction,  and  the  extremities,  which  are  free,  be  joined  by  a 
curve,  this  curve  must  be  of  such  a  nature  that  at  any  point  it 
may  be  at  right  angles  to  the  geodesic  curve  drawn  from  the 
common  point  to  that  point. 

335.  We  close  the  discussion  of  this  subject  by  the  con- 
sideration of  one  more  particular  geodesic  curve,  the  discus- 
sion of  which  is  not  without  interest,  and  appears  to  be  due 
to  Joachimstal. 


GEODESIC  CURVES.  399 


Problem  LIV. 

//  is  required  to  determine  the  nature  of  the  geodesic  curve 
drawn  upon  the  surface  of  a  spheroid. 

Let  the  equation  of  the  surface  be 

^  +  ¥  +  J^=''  W 

Hence  equations  (2),  Art.  333,  become 

.  :."  =  ^-,         f=y^.  (2) 

a  z  z 

Now  let  P  denote  the  perpendicular  from  the  centre  upon 
the  tangent  plane  to  the  surface  at  any  point  of  the  required 
curve,  and  D  the  semi-diameter  of  the  spheroid  drawn  parallel 
to  the  tangent  of  the  required  curve  through  the  same  point. 
Then  it  is  known  that 


and 


P'       a'  '^  b'^  b'       ^^ 


D'~  a'~^  b''^  b'  ~  "''' 


(3) 


Now  differentiating  v,  and  putting  for  x^'  and  /'  their  values 
from  (2),  we  have 

,       2b'  Ixx'   ,    yy'  ,  zz\    „       bWz" 

But  if  we  differentiate  (i)  twice,  and  in  the  result  substitute 
for  x"  and  y'^  their  values  from  (2),  observing  also  the  values 
of  u  and  v,  we  shall  obtain 

bhiz" 
v= --.  (5) 


400  CALCULUS  OF  VARLATIONS. 

Dividing  (4)  by  (5),  we  have 

v'  u' 

—  = ,        udv  +  vdu  =  o,  {6\ 

V  u  ^  ^ 

whence  uv  is  a  constant,  say  c. 

It  appears,  therefore,  from  the  first  and  third  members  of 
equations  (3)  that  the  geodesic  curve  must  be  of  such  a  nature 
as  will  always  render  PD  a  constant,  say  c'"^, 

336.  We  can  also  deduce  another  property  of  this  curve. 
Let  r  be  the  radius  of  curvature.  Then  it  is  well  known  that 
for  any  curve  in  space  we  have 

7  =  ^'"+/"+^'".  (7) 

Substituting  in  this  equation  for  x""^  and  y""^  their  values  from 
(2),  and  observing  also  the  value  of  u,  we  have 


But  from  (5)  we  have 


so  that 


I        uz"''b' 

r"          z" 

3.V 

e 

P' 

z'     ~  u'' 

-  D^' 

I 

v'        P' 

D' 

7 

r  — 

P 

(8) 


(9) 


But  we  have  PD  —  c'"^  and  D^  =  -p-^\  and  therefore  (9)  gives 

We  see,  then,  that  the  radius  of  curvature  for  the  geodesic 
curve  must  vary  inversely  as  the  cube  of  the  perpendicular 


BRACHISTOCHRONE  UPON  A    GIVEN  SURFACE.  4OI 

drawn  from  the  centre  of  the  spheroid  to  its  tangent  plane 
which  touches  the  geodesic  curve. 

337.  We  now  pass  to  another  problem  which  is  of  con- 
siderable interest,  following,  as  before,  the  guidance  mainly  of 
Prof.  Jellett. 

Problem  LV. 

A  particle  is  coinpelled  to  move  in  a  groove  upon  a  given  sur- 
face from  one  fixed  point  to  another^  being  urged  by  a  system 
of  forces  which  always  re?ider  Xdx  -\-  Ydy  -\-  Zdz  a  perfect  dif- 
ferential. Then  it  is  required  to  determine  the  nature  of  its  path 
in  order  that  it  may  move  from  the  first  point  to  the  second  in  a 
minimtun  ti^ne. 

Let  t  denote  the  time,  V  the  tangential  velocity,  and  ds  an 
element  of  the  required  path.  Then  it  is  evident  that  we  are 
to  determine  the  curve  which  will  minimize  the  expression 

But  we  have,  by  a  well-known  principle  of  mechanics, 

V  =  2f{Xdx  +  Ydy  -\-Zdz)=  f{x,  y,  z). 

Hence  we  conclude  that  the  present  problem  is  merely  an- 
other case  of  Prob.  LI  I.,  to  which  it  may  be  reduced  by  writ- 
ing-p=^. 

But  before  we  can  employ  any  of  the  formulae  obtained  in 
that  problem,  we  must  also  be  able  to  determine  the  values  of 
the  partial  differential  coefficients  of  v  with  respect  to  x,  y  and 
8*    Now  we  have 

-d{V'):=VdV=Xdx-\-Ydy^Zdz,  (i) 


402 


CALCULUS  OF  VARIATIONS. 


where  the  differentiation  is  total.  But  since  F,  and  conse- 
quently F^  is  a  function  of  x,  y  and  z,  the  partial  differential 
of  V^  with  respect  to  any  of  these  variables,  as  x,  is  the 
change  which  it  would  undergo  if  we  could  change  x  into 
X  -\-  dx,  the  other  variables  remaining  unaltered,  and  this 
change  is  given  in  (i)  by  making  dy  and  dz  zero.  Hence,  de- 
noting partial  differential  coefficients,  as  before,  by  literal 
suffixes,  we  have 


-{V%=  vv,  =  x, 


VVy  =   Y, 


VV,=  Z.} 


(2) 


Now  putting  for  v  its  value  — ,  we  find 


VV,           X 

Y 

z 

(3) 


Substituting  these  values  together  with  that  of  V  in  the  first 
of  equations  (i8).  Art.  328,  we  obtain 


V 


-  (Xcos  A  +  Fcos  ^  +  Zcos  C)\ 


(4) 


338.  Although  we  cannot  carry  the  solution  any  further 
while  the  problem  retains  its  present  general  form,  yet  we 
can  deduce  some  interesting  properties  belonging  to  this 
class  of  brachistochrone  curves. 

Let  R  denote  the  resultant  of  the  forces  at  any  point  of  the 
required  curve,  0  the  angle  made  by  the  osculating  plane  to 
the  curve  at  that  point  with  the  plane  of  that  normal  section 
which  contains  the  tangent  to  the  required  curve  at  that 


BRACHISTOCHRONE    UPON  A    GIVEN  SURFACE.  403 

point,  and  O  the  angle  which  R  makes  with  the  perpendicular 
to  the  plane  of  this  normal  section  erected  at  the  aforesaid 
point.     Then  we  know  that 

-S^cos'^^,         I-— ,==sm^,         —-—^——~,        (5) 

Now  the  aforesaid  perpendicular  to  the  plane  of  the  normal 
section  makes  angles  with  the  co-ordinate  planes  whose  co- 
sines are  numerically  equal  to  cos^,  cosB  and  cos  C.  Hence 
we  see  that 

(Xcos  A  +  Fcos  ^  +  Zcos  Cy  =  R'  cos'  0,  (6) 

Therefore  we  have 

sin'^      R'cos'O  V'sm^o 


F'sin^ 


=  R'qo^'0. 


=  R  cos  O. 


(7) 


339.  It  is  evident  that  the  members  of  the  last  equation 
may,  so  far  as  the  preceding  equations  are  concerned,  have 
contrary  signs,  and  we  must  therefore  next  justify  our  assump- 
tion that  they  should  be  taken  alike. 

Now  the  pressure  upon  the  curve  in  any  direction  is  equal 
to  the  sum  of  the  components,  in  that  direction,  of  the  result- 
ant and  of  the  centrifugal  force.  Moreover,  the  total  force  at 
any  point  may  be  resolved  into  three  :  the  first  normal  to  the 
surface,  which  is  destroyed  by  the  surface ;  the  second  along 
the  tangent  to  the  required  curve,  which  tends  to  produce 
acceleration  of  motion  ;  and  the  third  in  the  direction  of  that 
perpendicular  which  has  been  previously  mentioned,  and  thic 
component  would,  if  the  particle  were  constrained  to  move  in 

F'  . 
a  groove,  cause  pressure  against  its  side.     But  —y  is  the  cen- 


404  CALCULUS  OF  VARIATIONS. 

trifugal  force,  and  o  is  the  complement  of  the  angle  made  by  r^ 
with  the  aforesaid  perpendicular ;  so  that  the  members  of  the 
last  equation  equal  numerically  the  respective  components  of 
the  centrifugal  force  and  the  resultant  in  the  direction  of  this 
perpendicular.  Now  if  the  components  have  contrary  signs, 
then,  since  the  pressure  upon  the  side  of  the  groove  must 
equal  their  sum,  it  must  become  zero.  But  in  this  case  no 
groove  would  be  required  ;  the  motion  of  the  particle  upon 
the  surface  being  controlled  solely  by  the  given  system  of 
forces.  But  in  accordance  with  the  principle  of  minimum 
action,  the  path  of  the  particle  would,  under  the  present  sup- 
position, be  that  of  minimum  action  upon  the  given  surface 
with  the  given  forces ;  which  is  not  the  problem  we  wish  now 
to  discuss.  The  last  of  equations  (7)  is  therefore  correctly 
written  for  this  case. 

34-0.  We  see,  then,  from  the  last  of  equations  (7),  that  the 
required  curve  must  be  such  as  to  make  the  component  of 
the  centrifugal  force  perpendicular  to  the  plane  of  that  nor- 
mal section  which  contains  the  tangent  to  the  required  curve 
equal  to  the  component  of  the  resultant  in  the  same  direction. 

Again,  we  have 

Y^---^Rqq^0^2Rqo^G\  (8) 

which  equation  shows  that  if  a  particle  urged  by  a  system  of 
forces  move  on  a  given  surface  in  a  groove  of  such  a  form  as 
to  render  the  time  of  passing  from  one  fixed  point  to  another 
a  minimum,  the  pressure  upon  the  side  of  the  groove,  when 
the  particle  is  in  motion,  will  be  double  what  it  would  be  if 
the  particle  were  at  that  point  held  in  a  state  of  rest  and  still 
urged  by  the  same  forces. 

Again,  if  the  resultant  should  lie  in  the  plane  of  the  afore- 
said normal  section,  cos  O  will  vanish,  and  from  (6)  we  shall 
have 


MINIMUM  CURVE   OF  CONSTANT  CURVATURE.  405 

Xq,o^A^Yqo^B-\-Zqo^C—q, 
which,  in  (4),  gives 


=  o. 


Therefore  the  curve  in  this  case  must  be  a  geodesic  curve. 

341.  The  following  problem  is  also  from  the  work  of  Prof. 
Jellett,  and  its  complete  solution  appears  to  be  due  to  him, 
although  the  problem  itself  had  been  previously  discussed  by 
Delaunay. 

Problem  LVI. 

//  is  required  to  determijie  the  nature  of  the  curve  of  umiiniuni 
length  which  can  be  drawn  between  two  fixed  points  in  free  space, 
the  radius  of  curvature  of  the  curve  being  always  an  assigned 
constant. 

Let  ds  be  an  element  of  the  required  curve,  and  r  the 
radius  of  curvature,  which  is  a  constant.  Then,  adopt- 
ing here  also  the  arc  as  the  independent  variable,  we  are 
to  determine  the  curve  which  will  minimize  the  expression 

U  —  I  ^ds.  We  have  also,  in  order  that  we  may  be  able  to 
employ  the  method  of  Lagrange,  the  two  additional  equations 

:c'-  +y^  +  y^  =  I,         x'"  +/''  +  ^'"  =  ~  =  R\  (i) 

Therefore,  since  the  variation  of  W  can  give  only  the  terms 
ds^  —  ds^,  it  is  easy  to  see  that  by  following  the  method  of 
Lagrange,  /  and  /^  being  two  undetermined  quantities,  we 
shall  obtain  the  equations 

{/,x")"  -  {/x'Y  =  o,      {lyr  -  W)'  =  o. ) 

{i/r  -  (&')' = o.  ) 


406  CALCULUS  OF  VARIATIONS. 

Whence,  by  integration, 

(l^x")'  -Ix'  =  a^  x"i;  +  I/"  -  Ix', 

{lyy  -  ly'  -  b  =  y"i;  +  ly"  -  ly, 
{i/y  -iz'  ^  c  ^  z"i;  +  i^z'"  -  iz'. 


(3) 


Eliminating  /  between  the  first  and  second  of  these  equations, 
we  obtain 

{^y'jc"  -  x'y")  i;  +  {y'x'"  -  x'y'")  /,  =  ay'  -  bx'. 

This  equation  is  immediately  integrable,  giving 

I  J  {/x"  —  x'y")  =  ay  —  bx  -\-  f.  (4) 

In  like  manner,  eliminating  /  between  the  third  and  first,  and 
then  between  the  second  and  third  of  these  equations,  and  in- 
tegrating the  tw^o  resulting  equations,  we  have 

l^  {x'z'  —  z'x")  =^  ex  —  az  +/,,   ) 

\  (5) 

342.  Before  proceeding  further  we  must  consider  the 
mode  of  determining  the  constants  in  (4)  and  (5),  and  we  begin 
by  determining  /^  and  /.  For  this  purpose,  multiply  the  first 
of  equations  (3)  by  x'\  the  second  by  y",  the  third  by  z",  and 
add  the  products,  observing  that  equations  (i)  hold,  and  that 
hence 

jc'x"  +  yy  +  z'z"  =  o     and  x"x'"  +  yy  +  z"z'"  =  o.     (6) 

Then  we  have 

i^v/  =  «y'  +  ^y'  +  r.--,      R^/^^ax'  +  by  +  cz'  +  o-,   (7) 

Differentiating  the  first  of  equations  (6),  and  transposing,  we 
have 

x'x'" + yy  +  z'z'"  =  -  {x"'  +y''  +  z'")  =  -  r\     (8) 


MINIMUM   CURVE   OF  CONSTANT  CURVATURE.         407 

Now  multiply  the  first  of  equations  (3)  by  x' ,  the  second  by 
y,  the  third  by  z\  and  add  the  products,  observing  the  first 
of  equations  (i),  and  also  equation  (8).     Then  we  obtain 

l^  -  RH^-ax'  -  by'  -  cz'.  (9) 

Hence,  by  the  second  of  equations  (7),  we  have 

l=.g-2RH,  (10) 

343.  We  must  next  consider  the  terms  at  the  limits. 
Giving  merely  their  general  form,  these  are : 

Z  =  ^i-  + 1  Ix'  -  (I/')' \dx-^\ly'-  {lyy  \ 6y  -f ^ \ Iz'  -  {Iz")' \ dz 
j^l^S^:c"Sx'-^y"dy'^z"Sz'\^o.  (11) 

Now  suppose  the  extremities  of  the  required  curve  to  be 
fixed,  but  the  extreme  tangents  to  be  wholly  unrestricted. 
Then  it  is  evident,  first  of  all,  that  L^  and  L^  must  severally 
vanish. 

Now  consider  Z,,  and  take  first  those  terms  only  which 
contain  ds^,  Sx^,  dj/,  and  Sz^.  Then,  because  the  extreme 
points  are  fixed,  we  shall  have,  as  usual, 

^x^  —  —  x/ds^,         Sy^  =  —  y/ds^,         dz^  =  —  z/ds^,     (12) 

which  being  substituted  in  that  part  of  Zj,  having  first  written 

will  give,  by  employing  the  first  of  equations  (i)  and  (6),  and 
also  equation  (8), 

(i-/+R'/Xds,.  (13) 

Now  it  is  evident  that  we  could,  without  restricting  the 
extreme  tangents,  so  vary  the  arc  as  to  produce  no  change  in 
its  length,  in  which  case  ds^  would  vanish,  and  the  remaining 


408  CALCULUS  OF    VARLATLONS. 

part  of  L^  would  then  vanish  also.  Hence  we  see  that  the 
two  parts  of  L^  are  independent,  and  we  have 

L,ix"dx'  -\-/dy'  +  z''dz\  =  o.  (14) 

As  this  equation  can  be  satisfied  by  making  either  factor  zero, 
let  us  suppose  the  second  to  vanish.  Then,  although  Sx/,  (5y/ 
and  dz/  zltq  not  independent,  we  have,  from  the  first  of  equa- 
tions (i), 

x'dx'+/dy  +  z'dj2'  =  0;  (15) 

and  if,  by  this  equation,  we  ehminate  any  one  of  the  varia- 
tions, as  ^z/,  the  two  remaining  variations  may  be  regarded  as 
independent,  and  their  coefficients  be  equated  severally  to 
zero. 

Now  in  the  second  factor  of  (14)  first  eliminate  (^^/,  and 
equate  to  zero  the  coefficients  of  Sx/  and  dj// ;  then  eliminate 
(5j//,  and  equate  to  zero  those  oi^x/  and  <^z/.  Then  we  shall  ob- 
tain 

i^W  -  x'z"\  =  o,     {s'/'  -y'B"),  =  o,      {x'f  -y'x"\  =  o.  (i6) 

If  now  we  square  these  equations  and  add  them,  and  then  to 
the  sum  add  the  square  of  the  first  of  equations  (6),  we  shall 
obtain  a  result  which  may  be  written 

(^"+/"+o.  (^"+y^+o,  =  ^^  (17) 

the  last  member  following  from  equations  (i).  This  would 
make  the  radms  of  curvature  infinite  at  the  upper  limit;  and 
as  it  is  to  have  a  constant  value  throughout  [/,  the  required 
curve  would  become  a  right  line.  But  if  we  reject  this  solu- 
tion and  require  that  the  radius  of  curvature  shall  have  a  con- 
stant finite  value,  the- second  factor  of  (14)  cannot  vanish,  and 
/,,  must  vanish. 

Now  since  the  coefficient  of  ds^  in  (13)  must  also  vanish,  we 
see  that  /,  must  become  equal  to  unity.    These  values  make  ^ 


MINIMUM   CURVE   OF  CONSTANT  CURVATURE.  409 

in  equation  (10)  also  unity,  and  the  second  of  equations  (7)  be- 
comes 

l^FJ^ax'^by'  -\-cz'^\,  (18) 

344.  rt  is  evident  that  we  can  treat  L^  in  a  similar  man- 
ner, and  shall  obtain  like  results ;  so  that  we  have 

^/i==o.         ^,0  =  0,         /,  =  1,         /o==i.  (19) 

Now  since  the  position  of  the  origin  is  in  our  power,  assume 
it  at  the  lower  limit.  Then,  since  x^,  y^  and  z^  must  vanish, 
we  see  at  once  from  equations  (4)  and  (5)  that/,/  and/^ 
must  severally  vanish.  Then,  neglecting  /  /  and  /^,  multi- 
ply equations  (4)  and  (5)  by  z' ,  y'  and  x'  respectively,  and  add 
the  products.     Then  we  shall  find 

{ay  —  bx)z'  -\-  {ex  —  az)/  -j-  (bz  —  cy)x'  =  o 

=  a{yz'  -  zy')  +  b{zx'  -  xz')  +  c{xy'  -  yx').  (20) 

To  integrate  this  equation,  put  zrj  ior  y,  and  xv  ior  z.  Then 
(20)  becomes 

du      dv 

ail  —  b      av  —  c 

so  that,  by  integration, 

l{au  —  b)  =  l{av  —  c)-\-  c^=.  l{av  —  c) -\-  Ic^^  =  Ic^fyCiv  —  c). 

Now  putting  for  u  and  v  their  values,  removing  the  logarith- 
mic sign,  and  clearing  fractions,  we  have 

ay  —  bx  =  c"{az  —  ex),  (2ij 

which,  being  an  equation  of  the  first  degree  between  three 
variables,  is  the  equation  of  a  plane,  and,  containmg  no  con- 
stant term,  the  plane  passes  through  the  origin.     Now  the 


410  CALCULUS  OF    VARIATIONS. 

circle  is  the  only  plane  curve  of  constant  finite  curvature,  and 
this  must  therefore  constitute  the  solution  required. 

345«  But  it  is  easy  to  see  that  the  solution  just  obtained 
cannot  always  be  applicable.  For  suppose  the  assigned  value 
of  r  to  be  less  than  one  half  the  line  AB,  A  and  B  being  the 
two  fixed  points.  Then  the  circle  whose  radius  is  r  cannot 
pass  through  both  points,  so  that  we  are  led  to  expect  that  if 
there  can  be  any  solution  in  such  a  case,  it  must  be  discon- 
tinuous. 

Now  as  no  boundary  presents  itself  along  which  the  varia- 
tions of  X,  y  and  z  are  subject  to  any  other  restrictions  than 
those  which  are  imposed  by  equations  (i),  we  infer  that  the 
discontinuous  solution  can  consist  only  of  some  combination 
of  arcs  which  satisfy  equations  (2),  and  consequently  equations 
(3),  which  may  be  regarded  as  fundamental.  Still  it  is  evi- 
dent that  we  may,  *as  usual  in  cases  of  discontinuity,  suppose 
a,  b  and  c  to  have  each  different  values  for  the  different  points 
of  the  discontinuous  solution. 

But  in  the  present  case  these  constants  cannot  change 
their  values.  For  let  x^,  y^  and  z^  be  the  co-ordinates  of  the 
point  in  which  two  of  the  arcs  which  make  up  the  discontinu- 
ous solution  meet.  Then  the  part  of  ^^  without  the  integral 
sign  corresponding  to  this  point,  considered  as  being  on  the 
first  arc,  will  involve  only  dx^,  dy^,  Sz^,  dx^\  dy^  and  dz^\  For 
it  is  only  necessary  to  add  the  increment  ds  to  the  extreme 
limits  s^  and  s^,  as  the  only  reason  why  such  increments  are 
required  is  that  we  may  obtain  the  privilege  of  varying  the 
arc  in  the  most  general  manner,  which  would  require  an  in- 
crease or  decrease  in  its  length  as  a  whole.  Now  the  coeffi- 
cients of  6x^,  dy^  and  dz^  are  the  first  members  respectively  of 
equations  (3),  with  contrary  signs;  so  that,  denoting  this  part 
of  dUhy  Z2,  we  have 

L,=.-  adx,  -  bdy,  -  cSz,  +  l^lx"dx'  +  y"Sy'  +  z"dz'\.    (22) 


MINIMUM  CURVE   OF  CONSTANT  CURVATURE.         4II 

If  now  we  denote  by  x^,  y^  and  z^  the  co-ordinates  of  the  same 
points  considered  as  being  upon  the  second  arc,  we  shall  have 
at  that  point,  as  in  the  case  of  two  co-ordinates,  the  terms 
L^  —  Zg,  Z3  having  the  same  form  as  L^.  Hence  these  terms 
will  not  vanish  unless  a,  b  and  c  have  the  same  values  for 
each  arc. 

346.  It  appears,  then,  that  the  solution  must  consist  of 
some  combination  of  circular  arcs,  all  having  the  radius  r, 
and  situated  in  the  same  plane.  But  l^  must  vanish  at  the  ex- 
treme limits ;  and  we  see  from  (22).  that  to  make  L^  —  L^  vanish, 
we  must  also  make  /^^  and  l^^  severally  vanish,  or  must  have 

■^/  =  -^V>  7/= /a'.  ^/^-S"/,  /,,  =  /,3.  (23) 

We  see,  also,  from  (18),  that  when  the  first  three  of  equa- 
tions (23)  are  satisfied,  the  last  will  be  satisfied  also ;  so  that 
we  infer  that  the  arcs  are  also  to  be  so  placed  as  to  have  a 
common  tangent  at  their  point  of  meeting,  unless,  indeed,  we 
can  make  l^^  and  l^^  vanish  without  such  a  construction. 

Now  since  a,  b  and  c  are  unchangeable  throughout  the 
integral,  (4)  and  (5),  Avhich  are  derived  from  the  fundamental 
equations  (3),  must  also  hold,  as  must  equation  (18);  and  as 
the  arcs  must  lie  in  one  plane,  we  need  no  longer  employ  three 
co-ordinates.  Assuming,  therefore,  the  plane  of  the  arcs  as 
that  of  xy,  make  z  and  its  differentials  zero.  Then,  because 
l^  must  vanish  at  both  extreme  limits,  while  x^  =  o  and  y^  =  o, 
it  is  clear  that/,  /,  /^  and  c  must  vanish,  so  that  equations  (4), 
(5)  and  (18)  become  respectively 

l^iy'x"  -  x'y")  =  ay  -  bx,         l^  =  r\i -\-  ax'  +  by'),      (24) 

34-7,  It  appears,  then,  that  arcs  of  the  same  circle,  so 
joined  as  to  have  a  common  tangent,  will  give  at  least  one 
solution  of  the  problem,  provided  equations  (24)  are  satisfied 
throughout  the  entire  range  of  the  integration  ;  and  this  point 
we  next  proceed  to  consider. 


412  CALCULUS  OF  VARIATIONS. 

"Let  A  and  B  be  the  two  fixed  points,  and  suppose  we  take 
three  arcs,  A  CD,  DEF  and  FGB  ;  and  moreover,  since  the  ori- 
gin only  is  fixed,  it  being  at  A,  let  the  axis  of  x  take  the  direc- 
tion AB. 

E 

^  B 


Now  taking  first  the  arc  A  CD,  its  general  equation  may  be 
written 

(x~hyj^{y-kf=r'-  (25) 

where  h  and  k  are  the  co-ordinates  of  the  centre  ;  and  Ave  shall 
suppose  X  and  y  to  be  so  estimated  as  to  render  these  co- 
ordinates positive.     Differentiating  (25),  we  have 

{x  -  h)x'  +  (7  -  k)y'  =  o.  (26) 

Substituting  from  (26)  in  the  first  of  equations  (i),  we  easily 
find 

^'  ■=±{y-  k)R,         y  =^:{x-  k)R.  (27) 

Now  if  we  suppose  x  and  s  to  increase  together,  x'  is  always 
positive ;  and  y  ~  k  being  negative,  we  must  take  the  nega- 
tive sign.  But  the  arc  being  below  x,  y'  will  be  positive  or 
negative  according  as  x  -  h  is  positive  or  negative;  so  that 
for  It  we  take  the  positive  sign. 
We  therefore  have  for  this  arc 

x'-^-  R{y  _  k),  /  =  R{x  -  h\  \ 

X"  =.-Ry'^_  T^Y^r  ~h\  y"  =:  Rx'  =   _  R\y  _  k\   )     ^^^^ 

Substituting  these  values  in  equations  (24),  they  become 

-  Rl^  ^ay~  bx,         l^  =  r'\i+a{y~  k)R  +  b{x  -  h)R}.  (29) 


MINIMUM  CURVE   OF  CONSTANT  CURVATURE.  413 

Substituting  in  the  first  of  these  equations  the  value  of  l^  from 
the  second,  we  obtain 

r  -\-  ak  —  bh  ^^  o.  (30) 

Now  consider  the  arc  DBF,  and  let  H  and  K  be  the  co- 
ordinates of  its  centre.  Then,  proceeding  as  before,  we  shall 
find  that  we  must  now  reverse  the  signs  of  ^  and  y,  which  will 
leave  those  of  x^^  and  y  unchanged,  and  equations  (24)  will 
become 

ie/,  =  aj-  dx,        /^  =  r'\i+  a{y  -  K)  R  -  b{x  -  H)R\, 
Whence  we  obtain,  as  before, 

r-aK+bH  =  o.  (31) 

But  since  the  arcs  have  the  same  radius,  and  a  common  tan- 
gent at  D,  we  must  have  K  =z  —  k]  so  that  (30)  and  (31)  give 
b{h  -\-  H)  —  o,  an  impossible  equation  unless  b  vanish.  Under 
this  supposition,  the  second  of  equations  (24)  becomes 

l,^r'{l-{-ax').  (32) 

But  x\  being  always  positive,  has  the  same  value  at  D  as  at  H, 
and  therefore,  since  /^  must  vanish  at  the  latter  point,  it  will 
vanish  at  the  former  also. 

If,  on  the  other  hand,  we  had  required  for  the  ^ltcACB  the 
conditions  which  would  cause  /,  to  vanish  at  D,  as  well  as  at 
A,  we  would  have  found  it  necessary  to  make  b  vanish,  be- 
cause, while  the  value  of  x'  is  the  same  at  both  points,  those 
of  y  are  numerically  equal,  but  have  contrary  signs,  and 
therefore  the  second  of  equations  (24)  could  not  otherwise 
be  satisfied.  Then  equations  (30)  and  (31)  would  become 
r-\-ak  =  0  and  r  —  aK  =  o,  so  that  K  =  —  /b,  a.s  before. 

It  appears,  moreover,  from  (32),  that  if  /^  vanish  at  A  and 
D,  it  will  also  vanish  at  F;  and  that  if  we  had  taken  any  num- 


414  CALCULUS  OF  VARIATIONS. 

ber  of  arcs,  instead  of  three,  /^  would  vanish  at  each  point  of 
junction. 

348.  We  see,  then,  that  the  proposed  system  of  arcs  not 
only  gives  a  solution  which  satisfies  equations  (23),  but  it  is 
also  that  which  is  necessary  in  order  that  /^  may  vanish  at 
each  point  at  which  discontinuity  occurs,  so  that  we  have  no 
reason  to  expect  any  other  solution. 

But  as  we  may  take  as  many  arcs  as  we  please,  all  having 
the  assigned  radius,  it  is  evident  that  we  can  make  the  system 
differ  practically  in  no  respect  from  a  right  line,  which  was  a 
former  solution. 

349,  We  have  thus  far  supposed  that  the  curve  is  to  be 
drawn  between  two  fixed  points,  but  let  us  next  require  its 
extremities  to  be  confined  to  two  surfaces  whose  equations 
are  v  ^  o  and  F=  o,  z*  and  V  being  functions  of  x^  y  and  z 
only.  Then,  considering  the  upper  limit,  we  see  that  Z,  be- 
comes, by  the  aid  of  (3), 

L^-=  ds-  aSx-  bSy-  cdz,-\- l^,{^"dx'-\-y'dy-\-z''dz'),=^  o;  (33) 

and  since  /^^  vanishes,  we  have 

Zj  =  ds^  —  aSx^  —  bSy^  —  cSz^  =  o.  (34) 

Now  let  X,  +  [_Sx,\  y^  +  \6y;\  and  z,  +  [dz;\  be  the  co- 
ordinates of  the  point  in  w^hich  the  required  curve,  after 
having  been  varied,  meets  the  surface.  Then  we  have,  omit- 
ing  the  suffix  i, 

v,iSx-\  +  Vy[dy-]  +  v,[_Sz-]  =  O.  (35) 

But  we  have  in  this  case 

[dx]  =dx  +  x'ds,  [(5>]  =  6>  +  yds,  {Szl  =  dz-\-z'  ds ; 

30  that  (35)  becomes 

{v^dx  +  Vy  Sy  +  7'^  dz\  +  {y^  x'  J^Vyy'  +  v^  z'\  ds,  =  O.      (36) 


MINIMUM  CURVE  OF  CONSTANT  CURVATURE.  415 

Substituting  the  value  of  ds^  from  (34),  we  obtain 

+  V^y  +  b  {vxx'  +  Vyf  +  v^  ^OK^Ji 

+  S^^  -\-ciyxx'  +  e/^y  +  v^  z')\fiz,  =  o.  (37) 

We  may  now  regard  Sx^,  dy^  and  Sz^  as  independent,  and 
may  therefore  equate  their  coefficients  severally  to  zero.  Per- 
forming this  operation,  we  easily  deduce 


Vx 

a 


c 


(38) 


and  a  similar  equation  in  V  evidently  holds  for  the  lower  limit. 
Now  from  equations  (4)  and  (5),  since  /^  vanishes  at  both  lim- 
its, and  /,  fj  and  f^^  are  zero,  we  have 


ay^  —  bx^  =  o, 
cx^  —  az^  =  o, 
dz,  —  cy,  =  o. 


ay,  —  bx,  =  o,  1 
ex,  —  az,  —  o 
bz,  —  ^fo  =  o-  J 


(39) 


Whence,  by  subtraction,  we  deduce 


X.  —  X^ 


Therefore,  from  (38),  we  obtain 

i—^—]  =  (     ^^     ]  =  (    ^^     ]  > 


(40) 


(41) 


and  a  similar  equation  in  Ffor  the  lower  limit. 

These  equations  show  that  the  straight  line  joining  the  ex- 
tremities of  the  arc  must  be  normal  to  the  two  given  surfaces. 


4l6  CALCULUS  OF  VARL4TI0NS. 

350.  We  have  hitherto  supposed  the  extreme  tangents  of 
the  required  curve  to  be  wholly  unrestricted  ;  but  if  we  re- 
quire these  tangents  to  have  certain  assigned  directions,  it  is 
evident  that  the  preceding  figure  cannot  always  give  the  gen- 
eral solution  of  the  problem,  since  these  tangents  might  be  so 
assigned  as  not  to  lie  in  the  same  plane. 

It  is  shown  in  the  following  manner  by  Prof.  Todhunter,  in 
his  History  of  Variations,  Art.  156,  that  the  solution  in  such 
cases  will  sometimes  be  a  helix.  The  discontinuous  solution 
will  be  found  in  Art.  154. 

Since  dx' ,  Sy'  and  dz'  are  zero  at  both  limits,  it  is  no  longer 
certain  that  /^  will  vanish  at  either  limit.  Let  us  suppose, 
however,  that  the  conditions  relative  to  the  limits  are  such 
that  in  equations  (4)  and  (5),  a,  b,f^  and  f^^  vanish.  Then  the 
second  of  equations  (7)  becomes 

RH,  =  cz'-irg.  (42) 

i\lso,  the  terms  at  the  upper  limit  will  now  become 

Zj  =  ds^  —  cSz^  =  o ; 

and  the  extremities  of  the  curve  being  fixed,  dz^  =  —  z/ds^ ;  so 
that  we  have 

I  +  cz/  =  o.  (43) 

But  Z,  also  gives  rise  to  equation  (13),  so  that 

XV,  -  /,)  =  «,'. 

Hence  we  see  from  (42)  that^=  /^  and  from  (10)  that  Z^,  van. 
ishes,  and  then  from  (13)  that  /^  =  i  =z  o-;  so  that  (42)  becomes 

.    '  /^  =  rXi+czy  (44> 

Now  assume  x  =  h  cos  v,  y  ^^  h  sin  v,  and  z  =  kv.  Then 
we  easily  obtain  the  following  equations: 


MINIMUM  CURVE   OF  CONSTANT  CURVATURE.         417 


ds 


-  _  \/lf^^,  x'  ==  ^_- 


y= 


Vh'-^-M' 


Vh'-\-k' 


—  X 


,2     I     /.a'  -^  7.2 


From  (46)  and  the  second  of  equations  (i)  we  find 
Hence  (44)  becomes 


ck 
and  since  a,  b,  f^  and  f^^  are  zero,  equation  (4)  becomes 

ck      )      je 


^Ni+-^ 


while  either  of  equations  (5)  gives 

ck       )       k 


/. 


r^^i 


c. 


(45) 


{46) 


(47) 


(48) 


Substituting  for  r'  in  (49)  its  value,  that  equation  becomes 


(49) 


SO  that  we  have 


Whence 


k  VW^^  =  c{h' -^  k') 


' rip' 

VW^'  =  %-ck. 

k 

c  ^  VW+J' 
k         h'-kf 


(50) 


(51) 


41 8  'CALCULUS  OF  VARIATIONS. 

Next  substituting  the  value  of  r^  in  (48),  it  becomes 


-f=V/i  -^k  +ck=-^  =  If  —^r^'  (52) 

From  equations  (50)  and  (52)  we  see  that  we  cannot  have 
h  and  k  equal ;  but  with  this  exception  the  assumptions 
X  =:  h  cos  Vy  y  =:  h  sin  V  and  z  ^^  kv  will  satisfy  all  the  con- 
ditions of  the  question,  and  the  helix  will  therefore  be  the 
solution  required. 

351.  When  problems  of  relative  maj^ima  or  minima  are  to 
be  considered,  the  same  method  must  be  adopted  as  in  the 
case  of  two  co-ordinates ;  that  is,  we  multiply  the  integral 
which  is  to  remain  constant  bj'  a  constant,  say  a  ;  and  it  seems, 
therefore,  unnecessary  to  introduce  here  any  question  of  this 
class.  Indeed,  as  the  method  of  treatmg  all  the  problems 
which  belong  to  this  section,  whether  of  absolute  or  relative 
maxima  and  minima,  is  quite  uniform,  our  knowledge  of  the 
calculus  of  variations  would  not  be  materially  increased  by 
their  multiplication.  Moreover,  these  questions  generally 
lead  us  into  work  of  considerable  length,  and  rarely  afford  us 
any  solution  in  finite  terms,  and  are  therefore  somewhat 
wearisome.  We  shall  therefore  merely  state  two  or  three 
additional  problems  which  the  reader  will  find  in  the  work 
by  Prof.  Jellett,  or  in  the  more  recent  French  work,  Calcul  des 
Variations,  by  Moigno  and  Lindelof. 

(i)  To  draw  between  two  fixed  points  or  curves  upon  a 
given  surface  a  curve  which  will  maximize  or  minimize  the 
expression 

Ur^  r\v-^Vx')ds, 

V  and  V  being  functions  of  the  co-ordinates  x,  y  and  z  only. 

(2)  Two  fixed  pomts  on  a  surface  being  given,  and  a  curve 
connecting  them,  it  is  required  to  draw  between  these  points 
a  curve  of  given  length  such  that  the  portion  of  the  given 


GENERAL  REMARKS.  419 

surface  included  between  the  given  and  the  required  curve 
may  be  a  maximum. 

(3)  To  find  the  form  which  a  cord  resting  upon  a  given 
surface  must  assume  in  order  that  its  centre  of  gravity  may 
be  as  low  as  possible. 

362.  It  will  readily  appear  that  while  the  adoption  of  s  as 
the  independent  variable  often  presents  great  advantages  in 
the  discussion  of  the  terms  of  the  first  order,  it  is  exceedingly 
unfavorable  to  a  successful  examination  of  those  of  the  second 
order.  For,  in  the  first  place,  even  when  the  limiting  values 
of  X,  y,  2,  etc.,  are  fixed,  we  would  be  obliged  to  add  to  SU 
the  terms 

and  then  the  relations  between  ds  and  6x,  6y  and  S2  at  either 
limit,  which  we  have  previously  used,  and  which  are  true  to 
the  first  order  only,  must  be  replaced  by  more  accurate  equa- 
tions. In  the  use  of  these  more  accurate  equations,  certain 
terms  of  the  second  order  will  evidently  arise  at  the  limits ; 
and  as  we  may  only  equate  those  of  the  first  order  to  zero, 
these  terms  cannot  be  neglected,  but  must  be  added  to  those 
already  in  the  second  order,  thus  rendering  them  more  com- 
plicated. 

In  the  second  place,  when  we  are  obliged  to  use  the 
method  of  Lagrange,  we  must  render  that  method  true  to  the 
second  order,  which  we  have  not  hitherto  done.  To  accom^ 
plish  this,  w^hether  ;i:  or  ^  be  the  independent  variable,  we  first 
take  the  variation  of  U  to  the  second  order.  Then,  supposing 
the  connecting  equation  to  ht  f{x,y,  ^)  =  o  =/,  we  shall  have 

^f  =  fx^^  +  fy^y  -\-  fz^2 

+  t(/-^^'  +  ^fxy^^^y  +  fyy^/  +  etc.)  =:  O. 


420  CALCULUS  OF   VARIATIONS. 

Hence  we  may  write  //  Sfdx  =  o,  where  the  Umits  depend 

upon  the  independent  variable. 

Now  after  having  given  to  /  such  a  value  as  will  cause 

the  terms  of  the  first  order  to  vanish  after  //  d/dx  has    been 

added  to  ^U,  we  must  remember  that  the  variations  in  the 
terms  of  the  second  order  are  not  independent,  but  are  still 
connected  by  the  equation /=  o. 

If  then  these  terms  should  be  certainly  invariably  positive 
or  negative,  we  have  a  minimum  in  the  former  and  a  maxi- 
mum in  the  latter  case.  But  as  we  shall  generally  be  unable, 
if /be  a  differential  equation,  to  impose  this  restriction  in  any 
explicit  manner  upon  the  variations,  we  shall  not  usually  be 
successful  in  determining  the  sign  of  these  terms.  Of  course 
when  /  is  a  differential  equation,  its  variation  is  taken  to  the 
second  order,  as  already  explained. 

363.  In  the  discussion  of  problems  involving  three  co- 
ordinates, we  have,  according  to  our  usual  method,  ascribed 
no  variation  to  the  independent  variable,  whether  that  vari- 
able be  X  or  s.  But  it  is  quite  common  among  writers  to  vary 
the  independent  variable  also,  just  as  has  been  already  ex- 
plained for  problems  of  two  co-ordinates. 

Consider  first,  for  a  moment,  the  case  in  which  ;r  is  the 
independent  variable.  Here  we  follow  without  change  the 
reasoning  of  Art.  264  until  we  arrive  at  equation  (7),  after 
which  we  still  follow  the  article,  only  observing,  in  finding  the 

values  of     -^     and  dF,  that  V  is  now  a  function  of  x,^',/, 

.  .  .  .  z,  z' ,  .  .  .  .  Then,  having  obtained  the  longer  expression 
for  SU,  which  will  replace  equation  (10),  it  is  evident  that 
equations  (15),  Art.  265,  will  still  hold  true  to  the  first  order, 
and  that,  by  the  same  reasoning  for  x  and  z,  we  shall  have  the 
additional  equations 


GENERAL  REMARKS.  4^1 

(^^^  =  ^  +  z"  Sx,         dz"  =  ^^  +  z'^'dx,       etc. ;     (i) 
dx  ax-  ^  ' 

where  cso^  ^^  Sz  —  z'Sx.  Therefore,  proceeding  as  in  Art.  266, 
we  shall  obtain,  instead  of  equation  (17),  an  expression  for  dU 
identical  in  form  with  that  which  would  result  from  ascribing 
no  variation  to  x,  except  that  oo  and  00^  will  replace  dy  and  ^z. 

Moreover,  since,  to  the  first  order,  gd  and  cs^  equal  dy  and 
dz  in  the  other  method,  we  see  that  whatever  relations  may 
hold  between  these  variations  when  the  ordinary  method  is 
employed  must  hold  also  between  od  and  (^  when  x  is  varied, 
so  that,  as  in  the  case  of  two  co-ordinates,  the  same  general 
equations  will  be  obtained  by  either  method,  and  it  will  be 
found  also  that  the  same  equations  at  the  limits  can  be  estab- 
lished by  either  method. 

Next,  when  s  is  the  independent  variable,  we  proceed  as  in 
Art.  296,  merely  observing,  in  finding  the  values  of  v'  and  Sv^ 
that  V  is  now  a  function  of  s  and  x,  y  and  z  with  their  differen- 
tial coefficients  with  respect  to  s.  Moreover,  we  shall  have,  in 
addition  to  equations  (7)  of  that  article,  which  will  still  hold 
true  to  the  first  order,  the  equations 

dz'  =  (GD^y  +  z"Ss,         6z"  =  {oD^y  +  z'^'Ss,        etc.,       (2) 

where  gd^  r=  Sz  —  z'Ss. 

Hence,  as  in  the  case  of  two  co-ordinates  only,  we  shall 
find  that  d^will  take  the  same  form  as  if  we  had  ascribed  no 
variations  to  s,  except  that  go^,  gdv  and  gd^  will  take  the  place 
of  6xj  6y  and  dz  respectively. 


CHAPTER   III. 

MAXIMA    AND    MINIMA    OF    MULTIPLE    INTEGRALS. 


Section  I. 

CASE  IN  WHICH  U  IS  A  DOUBLE  INTEGRAL;    THE  LIMITING 
VALUES  OF  X,    V,  Z,  ETC,  BEING  FIXED. 

Problem  LVII. 

354.  Suppose  we  require  the  form  of  the  surface  of  least  area 
terminated  in  all  directions  by  a  certain  fixed  and  closed  linear 
boundary. 

If  this  boundary  were  a  plane  curve  or  any  linear  figure 
situated  entirely  in  the  same  plane,  the  required  surface  would 
of  course  be  itself  plane.  But  we  here  wish  that  the  bound- 
ing frame  or  edge  may  have  any  assigned  form  whatever,  not 
inconsistent  with  the  condition  that  it  shall  be  closed. 

Suppose,  then,  the  required  minimum  surface  to  have  been 
obtained,  and  call  it  the  required  surface,  and  suppose  we 
take  any  other  surface  having  a  common  edge  with  the  first, 
and  call  this  the  derived  surface.  Then  it  will  appear,  as 
in  Prob.  I.,  that  to  prove  the  required  surface  to  be  that 
of  least  area  we  must,  in  the  first  place,  assume  that  the 
derived  surface  with  which  its  area  is  compared  differs  from 
it  in  form  infinitesimally  only.  Then  if  the  surface  found 
have  a  less  area  than  any  such  derived  surface,  it  will  be  a 


SURFACE    OF  MINIMUM  AREA.  423 

minimum,  that  term  being  used  in  the  technical  sense  already 
explained,  and  it  will  then  be  in  order,  in  discussing  the  least 
surface,  to  consider  whether  there  may  be  any  other  minima. 
We  shall  then  at  present  discuss  only  the  problem  of  find- 
ing the  minimum  surface. 

356.  Now  let  x,  y  and  z  be  the  co-ordinates  of  any  point  of 
the  required  surface,  and  suppose  four  indefinite  planes — two 
parallel  to  the  plane  of  yz,  and  two  parallel  to  that  of  xz, 
the  distance  between  the  former  two  being  dx,  and  between 
the  latter  two  dy.  Then,  denoting  by  ds  an  element  of  the 
surface  intercepted  at  any  point  by  any  four  planes  drawn  as 
above,  it  is  known  that  we  shall  have 


ds^Vi  -^z"-^z;dydx, 

where  accents  above  denote  total  differentiation  with  respect 
to  Xy  and  those  below  the  same  with  respect  to  y.  See  De 
Morgan's  Diff.  and  Integ.  Calc,  p.  444.  Therefore,  designating 
the  surface  whose  area  is  to  become  a  minimum  by  U,  we  have 
to  minimize  the  expression 

U=         /      Vi^z'-'-^z'dydx^  /      /     Vdydx,       (i) 

356.  It  is  essential  that  we  should  here  recall  from  the 
theory  of  double  integration  a  clear  conception  of  the  precise 
meaning  of  equation  (i).  Suppose,  then,  the  entire  surface  to 
be  divided  into  strips  by  planes  parallel  to  that  of  yz,  the 
distance  between  these  planes  being  dx.  Then,  the  area  in 
question  equals  the  sum  of  these  strips,  while  that  of  any  strip 
is  itself  equal  to  the  sum  of  the  elemental  areas  intercepted 
on  it  by  successive  planes  parallel  to  that  of  xz,  and  sepa- 
rated by  the  distance  dy. 

To  effect  this  latter  summation,  which  we  shall  always 
suppose  to  have  been  first  accomplished,  we  must  imagme 


424  CALCULUS  OF  VARIATIONS. 

the  value  of  V  to  have  been  obtained  from  the  general  equa- 
tion of  the  surface,  thus  rendering  Fsome  function  of  x  and 
y  only,  since  z  is  some  function  of  x  and  y ;  and  then,  as  x  and 
dx  will  have  the  same  value  for  every  element  of  the  same 
strip,  while  y  will  vary,  we  must  integrate  the  expression 
Vdydx  under  the  supposition  that  x  and  dx  are  constants. 
But  since  the  summation  must  extend  throughout  any  strip 
which  we  wish  to  consider,  if  we  denote  by  y^  and  y^  the 
values  of  y  at  its  extremities,  the  area  of  any  strip  will  evi- 

dently  be  given  by  the  expression  /      Vdydx,  x  and  dx  being 

treated  as  constants.     But  because  V  was  made  a  function  of 

X  and  J  ov\j,J  Vdy  will  be  a  function  of  these  quantities ;  and 

since  for  any  particular  strip  y^  and  y^  will  certainly  be  func- 
tions of  X  only,  and  perhaps  constants,  if  we  put  5  for  the  area 
of  any  strip,  we  may  write 

5  =  f{x)  dx  =  fdx,  (2) 

Now  to  effect  the  summation  of  the  strips,  which  is  always 
the  latter  process,  we  suppose  the  edge  or  contour  of  the  sur- 
face, when  it  has  been  projected  upon  the  plane  of  xy,  to 
form  a  curve  capable  of  being  expressed  by  the  equation 
y  —  Fix),  which  curve  we  shall  call  Xh^  projected  contour.  Then 
equation  (2),  which  was  before  true  for  any  strip,  becomes  so 
for  every  strip.  Hence  we  need  no  longer  regard  x  as  con- 
stant ;  and  integrating  from  x^  to  x^,  where  x^  and  x^  denote 
the  extreme  abscissas  of  the  surface,  or  rather  of  the  projected 
contour,  we  shall  obtain  the  entire  surface  U. 

357.  Hitherto  we  have  usually  employed  the  suffixes  o,  i, 
etc.,  to  denote*  what  the  quantity  to  which  they  are  applied 
will  become  when  the  independent  variable  receives  a  partic- 
ular value.  Now  because,  in  the  discussion  of  curves,  whether 
situated  in  space  or  not,  we  have  but  one  independent  vari- 


SURFACE   OF  MINIMUM  AREA.  425 

able,  ;ir  or  J  or  some  other,  this  method  is  satisfactory.  But  in 
problems  relative  to  surfaces,  where  no  curve  is  traced,  x  and 
y  are  evidently  entirely  independent,  so  that  the  substitution 
of  a  particular  value  of  one  variable  does  not  necessitate  the 
substitution  of  any  particular  value  of  the  other,  as  it  would  if 
we  were  discussing  a  curve.  It  is,  therefore,  important  that 
we  should  be  able  to  specify  just  what  substitutions  of  each 
variable  are  to  be  made  in  any  function,  which  cannot  be  con- 
veniently done  by  suffixes,  particularly  when  we  come  to  in- 
tegrals of  the  third  or  higher  order,  involving  three  or  more 
independent  variables. 

These  substitutions  are  indicated  in  the  following  simple 
manner.  Let  x,  y,  z,  etc.,  be  any  quantities  whatever,  and  let 
F  be  any  function  of  these  quantities.  Then  when  we  put  for 
any  of  these  quantities  a  particular  value,  as  x^  for  x,  we  write 

i%  it  being  always  understood  that  a  suffixed  quantity  is  sub- 
stituted for  the  unsuffixed  one  of  the  same  name.  Also,  if  it 
be  necessary  to  denote  that  x^  has  been  substituted  for  x  and 

y^  for  y,  we  write  the  new  function  thus,         I    F^  where  we 

shall  always  suppose  y^  to  have  been  first  substituted  for  y, 
after  which  x^  is  substituted  for  x  in  the  resulting  function. 

Again,  suppose  /  to  be  a  function  integrable  with  respect 
to  X,  its  general  integral  being  F,     Then  we  may  write 

J.Jdx  =  F,-F,=  l    F-  I    F 

Now  as  we  shall  often  have  expressions  of  a  similar  form  aris- 
ing from  definite  integration,  we  write  the  last  equation  thus, 

/    fdx  =  /^    F,  where  it  will  be  always  signified  that  we  are 

to  substitute  successivel}^  the  upper  and  lower  suffixed  for  the 
corresponding  unsuffixed  quantity,  and  then  subtract  the  sec- 
ond result  from  the  first. 


426  CALCULUS  OF   VARIATIONS. 

Extending  this  principle  still  further,  j^    j     F  will  denote 

the  following  operations :  tirst,  that  we  must  substitute  suc- 
cessively j^i  and  Jo  for  jK,  and  subtract  the  second  result  from 
the  first ;  and  second,  that  in  the  result  we  must  substitute  x^ 
and  x^  for  x^  and  subtract  as  before.     Thus  we  shall  have 

F=         \         F-         F 

Ixq     lyo  Ixq      )  I  / 

/iCi    /  ?/i  Ix^    ly^  Ixa    lyi  /a-o     /2/o 

=  1    I    F~l    I    P-l    I    F^l     I    F-  (3) 

358.  The  idea  of  employing  a  sign  to  denote  substitution 
is  due  to  M.  Sarrus,  who  calls  it  the  sign  pf  substitution,  a 
name  which  we  shall  retain ;  and  it  seems  probable  that,  as 
Prof.  Todhunter  has  remarked,  since  Lagrange  introduced  his 
symbol  ^,  nothing  has  been  suggested  which,  is  of  such  service 
to  the  calculus  of  variations  as  this  sign.  But  the  sign  and 
the  method  of  employing  it  were  subsequently  modified  by 
Cauchy,  whose  method  we  substantially  follow,  as  exhibited 
in  the  Calcid  des  Variations  by  Moigno  and  Lindelof. 

359.  As  an  illustration  of  the  preceding  discussion,  let  us 
suppose  the  given  surface  to  be  spherical,  taking  the  origin  at 
its  centre,  and  considering  only  some  portion  of  the  upper 
hemisphere,  whose  edge  or  contour  is  to  have  any  form  we 
please.  We  may  notice  that  z'  is  the  partial  differential  coef- 
ficient of  z  with  respect  to  x,  and  is  obtained  therefore  from 
the  equation  of  the  surface  by  regarding  y  as  constant ;  and 
similarly  x  must  be  constant  in  obtaining  z^.  The  equation  of 
the  sphere  is  x"  ^ y"  -^  z"  =  r".     Whence 


z'  = 


X                     —  X 

-I 

2        ^r'-x'-f 

z 

Vi+z-  +  z;=---^^—r—: 

Vr'-x"'-/ 


SURFACE   OF  MINIMUM  AREA,  427 

SO  that  (i)  becomes 

jj  __     n^x   rvi       rdydx 

~  tAo  Jyo  vp  —  x^  —y 

But  regarding  x  as  constant,  we  have 

/-— ^-^^rsin-      J^^  +  c: 

and  the  definite  integral  may  be  written 


S  =  /    rsm-' — ^ 


'yo  |/^.  _  ^. 


dx. 


Thus  we  see  that  5  does  not  contain  z,  and  is  also  indepen- 
dent of  the  general  values  of  y,  but  may  still  be  some  function 
of  X. 

Now  if  we  wish  to  denote  the  area  of  any  particular  strip 
for  which  x  =  Xa,  we  have  only  to  write 


/      /    r  sm  -   —  -^         dx. 

I      Iv^  ^/r"  -  x' 


To  complete  the  integration,  let  us  require  all  the  surface 
for  which  neither  x,  y  nor  2  shall  become  negative.  Then  we 
shall  have 

y.  =  0,  y,  =  Vr'  —  x\  S= ,     U=:  / =  /     ; 

and  since  the  entire  eighth  of  the  sphere  is  required,  Xa=  o 

and  x^  =  r,  and  [/  — 

2 

360.  Returning  now  to  our  original  problem,  we  see  that 
we  can  pass  from  any  given  surface  to  any  other  differing 
from  it  infinitesimally  in  form,  and  having  a  common  edge,  by 


428  CALCULUS  OF    VARIATIONS. 

giving  to  z  suitable  infinitesimal  increments  throughout  the 
surface,  the  values  of  both  x  and  y  undergoing  no  change  ;  and 
as  dz  indicates  the  change  which  z  undergoes  when  we  pass 
from  one  point  to  its  consecutive  on  the  same  surface,  we  des- 
ignate the  new  increments,  as  before,  by  Sz.  Moreover,  we 
can  also,  without  varying  x  or  jj/,  obtain  the  derived  surface  by 
giving  infinitesimal  variations  to  z'  and  z^,  which  are  the  tan- 
gents of  the  angles  made  with  the  plane  of  xy  by  those  two 
edges  of  any  elemental  area  which  meet  at  the  point  x^  y,  z. 

If  now  we  denote  by  (^^  the  change  in  area  which  the  en- 
tire surface  will  undergo  when  z,  z'  and  z^  receive  infinitesimal 
variations,  the  required  surface  must  evidently  be  such  as  to 
render  df/ negative.  But  as  we  cannot  express  ^in  any  more 
explicit  form  than  that  given  in  (i),  and  as  we  must  compare 
the  required  surface  with  such  as  can  be  derived  by  infinites- 
imal changes  in  its  form,  Ave  are  compelled  to  seek  the  varia- 
tion of  the  double  integral  in  (i)  in  order  to  determine  what 
conditions  will  render  the  variation  negative. 

361.  In  order  to  consider  the  subject  more  generally,  let 
us  assume  the  equation 

where  Fis  any  function  of  Xy  y,  z,  z'  and  z^,  the  limiting  values 
of  X,  y  and  z  being  fixed ;  and  let  us,  for  convenience,  write 
z'  =/,  ^^  =  q^  z"  =  r,  zl  —  s  and  z^^  =  /.  Then  if  we  change 
z  into  z  -\-'^z,  p  into  p  -\- ^p  and  g  into  g  +  Sg,  x,  dx,  y  and  dy 
remaining  unaltered,  and  denote  hy  SU  and  dV  the  changes 
which  U  and  V  will  undergo,  we  shall  have 

=  rrVdydx-\-rr,Vdydx. 


SURFACE   OF  MINIMUM  AREA,  429 

Whence,  from  (i), 

We  have  now  merely  to  determine  (^Fby  Taylor's  Theorem, 
which,  since  x  and  y  undergo  no  change,  will  give 

+  Vy^sp'^  2 1\  Sz  6q  +  2  Fp,  Sp  6q  +  V^^  Sq^  [  +  etc. ;     (3) 

where  the  etc.  denotes  terms  of  the  third  and  higher  orders, 
and  the  differentials  of  V  are  all  partial. 

362.  Now  denoting  by  vS  the  terms  of  the  second  order  in 
S  V,  with  the  exception  of  the  -,  we  shall  have 

^^=  £?fy''^  ^''^'+  V,,SpJ^V^Sq\dydx 

+  2X.  X  Sdydx-^^^o.  (4) 


If  now  we  require  that  U  shall  become  either  a  maximum 
or  a  minimum,  it  will,  since  Sz^  Sp  and  6q  are  entirely  in  our 
power  and  may  have  either  sign,  appear,  by  precisely  the 
same  reasoning  as  in  the  case  of  single  integrals,  that  the 
first  integral  in  (4)  must  vanish,  while  the  second  must  become 
invariably  positive  for  a  minimum  and  negative  for  a  maximum. 

Now  we  must  observe  that  x  and  y  are  completely  inde- 
pendent, and  that  z'  and  z"  or  /  and  r  are  the  differential  co- 
efficients of  z  with  respect  to  x,  y  being  regarded  as  constant ; 


430  CALCULUS  OF    VARLATIONS. 

that  is,  in  finding  them,  we  regard  -S'  as  a  function  of  x  only, 
and  constants,  among  which  we  reckon  y.  Or  we  may  regard 
z  as  the  ordinate  of  the  curve  made  by  the  intersection  of  the 
required  surface  with  a  plane  parallel  to  that  of  xz  at  the 
distance  y.  Similarly,  z^  and  z^^  or  q  and  t  are  the  differen- 
tials of  z  with  respect  to  y,  x  being  constant ;  that  is,  z  may 
now  be  regarded  as  the  ordinate  of  the  section  cut  by  a  plane 
at  right  angles  to  the  first,  and  at  the  distance  x  from  the 
plane  of  yz.  Therefore,  as  x  and  y  receive  no  variation,  we 
we  must  have,  as  heretofore, 

dz'     or     Sp  —  — — ,         8z"     or     Sr  —  —^-r-, 
ax  ax 

dSz  .  .^       d^Sz 

dz^     or     dq  =  ——,         oz.,     or     6t  = 


2      ' 


dy  '  ''  dy 

and  these  equations,  which  are  exact,  may  be  used  in  any 
manner  we  find  convenient  in  transforming  SU. 

363,  Considering  for  the  present  the  terms  of  the  first 
order  only,  we  have 

<^^=/"X"5  F,fo  +  rp(J/+  V,SqUfydx  =  o.  (5) 


But  without  entering  upon  any  general  discussion  of  the  con- 
ditions which  must  hold  in  order  that  (5)  may  be  satisfied,  let 
us  return  to  our  original  problem.     Here 


V=Vi+f  +  ^\         V,  =  o, 

K= 

J                      V  -           ^ 

SURFACE    OF  MINIMUM  AREA.  43 1 

SO  that  (5)  gives 

Now  we  cannot  assert  that  every  element  of  this  integral 
must  vanish,  because  we  have  also  required  that  the  edges  of 
the  surface  shall  be  fixed — that  is,  that  dz,  for  all  points  of  the 
edge  or  contour,  shall  vanish — and  this  condition  has  not  yet 
been  imposed  upon  dU.  Indeed,  there  is  an  analogy  between 
the  present  problem  and  Prob.  I.  For  in  Prob.  I.  we  were  to 
connect  two  fixed  points  by  a  line  of  minimum  length,  requir- 
ing us  to  minimize  a  single  integral ;  while  in  the  present  prob- 
lem we  are  to  connect  an  infinite  number  of  fixed  points, 
forming  the  given  contour,  by  a  surface  of  minimum  area,  re- 
quiring thereby  the  minimizing  of  a  double  integral. 

364-.  The  condition  just  mentioned  may  be  imposed  some- 
what as  in  Prob.  I.     For  we  have 


dqdy  dx 


^0     ^'Vo      |/i_|_^2_|_^2 

Szdx 


'^0    lyo     ^j  -f/-(-^^ 

If  now  we  remember  that  for  any  abscissa  x,  y^  and  y^  are  the 
two  ordinates  of  the  projected  contour  corresponding  to  this 
abscissa,  we  shall  see  that  the  ^'s  corresponding  to  y^  and  y^  re- 
late to  the  edge  or  contour  only  of  the  required  surface,  and 
that  therefore  every  dz  in  the  single  integral  in  (7)  must  van- 
ish, causing  the  integral  itself  to  vanish. 

Now  since  we  may  adopt  either  order  of  integration  in  a 


432  CALCULUS  OF  VARIATIONS. 

double  definite  integral  without  affecting  its  value,  we  may 
write 


_       P_ 


^^  dy 
_  p  l>y,d  p_ ^^      ^^_ 

Here  we  regard  y  as  the  independent  variable  in  the  equation 
of  the  projected  contour^  so  that  x^  and  x^  are  always  ordinates 
of  this  contour,  y  being  the  abscissa.  Hence,  as  before,  every 
Sz  in  the  single  integral  of  the  last  equation  refers  to  some 
portion  of  the  contour  only,  and  must  vanish. 
Hence,  finally,  we  must  have 

365.  It  is  here  necessary  to  notice  two  points. 

First.  It  will  be  seen  that  the  form  in  which  the  terms 
under  the  sign  of  single  integration— which  terms  are  the 
terms  at  the  Hmits  in  this  problem— have  been  left  is  incongru- 
ous, inasmuch  as  we  do  not  retain  the  same  independent  vari- 
able throughout.  But  our  only  object  at  present  is  to  show 
that  when  the  contour  is  fixed  the  terms  at  the  limits  will 
vanish.  Indeed,  the  arrangement  of  these  terms  in  the  case 
of  multiple  integrals,  so  as  to  enable  us  to  discuss  with  any- 
thing like  generality  the  conditions  which  must  hold  at  the 


SURFACE   OF  MINIMUM  AREA.  433 

limits,  has  proved  to  mathematicians  one  of  the  most  difficult 
points  connected  with  the  calculus  of  variations.  For  although 
this  subject  had  more  or  less  occupied  the  attention  of  Gauss, 
Poisson,  Ostrogradsky,  Jacobi  and  Delaunay,  the  last  of 
whom  has  been  followed  by  Prof.  Jellett,  it  remained  for  M. 
Sarrus  to  present  a  method  of  treatment  which  has  the  merit 
of  being  systematic  and  general,  and  is  perhaps  as  nearly  f»er- 
fect  as  the  nature  of  the  subject  will  permit. 

Second.  The  two  differentials  in  (9)  denote  the  entire 
change  produced  in  the  first  fraction  when  we  change  x  into 
X  +  dx,  and  in  the  second  when  we  change  y  into  y  -\-  dy,  p 
and  q  being  variable  both  for  changes  in  x  and  y,  so  that,  with 
respect  to  x  or  y  only,  these  differentials  may  be  said  to  be 
total.  Such  differentials  are,  however,  called  partial,  since 
they  denote  the  change  incident  upon  an  alteration  in  one  in- 
dependent variable  only,  while  there  are  two  which  might  be 
varied. 

366.  Now  the  two  single  integrals  in  (7)  and  (8),  taken  to- 
gether, certainly  involve  every  dz  for  the  contour,  which 
would  not  perhaps  be  true  of  the  first  integral  alone  if  a  por- 
tion of  the  projected  contour  should  be  a  right  line  perpen- 
dicular to  the  axis  of  x,  nor  of  the  second  if  a  portion  of  the 
projected  contour  should  be  a  right  line  perpendicular  to  the 
axis  of  J.  Hence  in  (9)  the  condition  that  Sz  shall  be  zero 
throughout  the  entire  contour  has  been  imposed. 

Now  as  the  sign  of  Sz  for  every  point  of  the  required  sur- 
face is  wholly  within  our  power,  and  its  value  is  subject  to  no 
other  restrictions  than  that  it  shall  be  infinitesimal,  and  shall 
render  dp  and  Sq  also  infinitesimal,  it  will  appear,  as  hitherto, 
that  we  can  only  satisfy  (9)  by  equating  M  to  zero,  so  that  we 
shall  have 

A^--^^-^-^±-—i-_-^^-M=o.        (10) 
dx  Vi-\-p'-\-q'     dy  4/1+/  +  ?" 


434  CALCULUS  OF  VARIATIONS. 

Performing  the  differentiation  indicated  in  the  last  equation, 
observing  that  p^  —  q'  —  s,  we  have,  after  reducing  to  a  com- 
mon denominator, 

This  expression,  which  is  a  partial  differential  equation  of 
the  second  order,  is  known  to  indicate  that  the  required  sur- 
face must  be  of  such  a  nature  that  at  every  point  the  principal 
radii  of  curvature  may  be  equal  and  taken  with  a  contrary 
sign,  so  that  their  sum  may  be  always  zero.  Moreover,  equa- 
tion (lo),  which  is  the  fundamental  equation,  would  evidently 
be  satisfied  by  a  plane,  since  /  and  q  would  then  become  con- 
stant. This  could  not,  however,  as  we  have  already  shown, 
be  the  general  solution,  because,  if  the  given  contour  were  not 
a  plane  figure  it  would  not  be  possible  to  make  a  plane  sur- 
face fulfil  all  the  conditions  at  the  limits ;  that  is,  to  pass 
through  every  point  of  the  given  contour.  But  we  shall 
resume  the  consideration  of  (ii)  presently. 

367.  Assuming  the  required  surface  in  any  particular  case 
to  have  been  determined,  let  us  now  examine  the  sign  of  the 
terms  of  the  second  order.  Since  c  does  not  enter  V  expli- 
citly, we  have,  from  (2)  and  (4), 

5  =  Vpp<Sf  +  2  VpqSp  dq  +  Vqq  6q' 

_{l+q^)Sf-2pqSpSqJr{l-\-p'')^g' 
(1+/  +  ^^)' 

Whence,  since  the  terms  of  the  first  order  vanish,  we  may 
write 

^^-2J^.Jy,    JJ^fj^g'), dydx,         (12) 


SURFACE   OF  MINIMUM  AREA,  435 

which  shows  that  dU  \^  invariably  positive,  since  every  ele- 
ment of  the  double  integral  is  essentially  so. 

We  see,  therefore,  without  solving  (lo),  that  so  long  as  the 
contour  of  the  required  surface  is  to  be  fixed,  any  surface 
which  satisfies  (lo)  or  (ii),  and  can  also  pass  through  every 
point  of  the  given  contour,  will  possess  a  minimum  area. 
We  should  not,  however,  say  that  the  surface  thus  found  is 
necessarily  that  of  the  least  area.  For  although  this  may  be 
true  in  the  present  problem,  the  method  of  variations  does 
not  of  itself  warrant  the  assertion.  This  will  at  once  appear 
if  we  remember  that  the  calculus  of  variations  permits  us  to 
compare  the  required  primitive  surface  with  such  derived 
surfaces  only  as  differ  from  it  infinitesimally  in  form  ;  and  we 
cannot,  therefore,  be  certain  that  there  might  not  be  some 
other  minimum  surface  whose  area,  being  less,  might  itself 
be  the  least  possible. 

Moreover,  since  ^p  and  ^q  must  be  infinitesimal,  we  are 
not  permitted  to  consider  any  step-shaped  surface ;  and  one 
of  these  might,  perhaps,  be  that  of  least  area.  In  fact,  it  will 
appear  that,  theoretically  at  least,  the  distinction  between 
maxima  and  minima,  and  greatest  and  least,  values  must  hold 
equally  whether  the  integral  be  single  or  double. 

368.  Let  us  now  return  to  the  terms  of  the  first  order. 
It  is  easy  to  see  that  had  the  equation  been 


where  V  contained  x,  y,  z,  p  and  q,  the  same  reasoning  by 
which  we  obtained  (lo)  would  have  given  us  the  equation 

J/=r,-(F^)'-(r,)=o;  (.3) 

and  from  the  case  of  single  integrals  we  would  naturally  infer, 
what  we  shall  presently  show,  that  this  equation  will  be  true 


43^  CALCULUS  OF  VARIATIONS. 

independently  of  any  conditions  which  may  be  required  to 
hold  at  the  limits. 

Indeed,  this  fundamental  equation  appears  to  present  itself 
naturally,  and  to  have  been  obtained  almost  as  soon  as  the  sub- 
ject was  discussed  ;  while  no  subsequent  researches  have  given 
us  any  other  equation.  Now  (13)  will  be  what  is  known  as  a 
partial  differential  equation  of  the  second  order,  the  variables 
X  and  y  being  entirely  independent,  and  z  being  supposed  to 
be  some  function  of  these  variables.  But  the  theory  of  such 
equations  is  still  imperfect,  it  being  uncertain  even  that  every 
partial  differential  equation  of  the  second  order  has  any  solu- 
tion at  all ;  so  that  we  are  very  rarely  able  to  obtain  the  com- 
plete  integral  of  the  equation  J/  =  o,  or  indeed  to  obtain  any 
solution  whatever  in  finite  terms. 

We  know,  however,  that  when  a  partial  differential  equa- 
tion of  any  order  can  be  integrated  completely,  the  integra- 
tion will  introduce  certain  arbitrary  functions  instead  of  the 
ordinary  arbitrary  constants,  and  that,  however  the  solution 
be  obtained,  the  number  of  these  arbitrary  functions  will  not 
exceed  that  of  the  order  of  the  partial  differential  equation. 

369.  According  to  Moigno,  the  integral  of  equation  (10) 
was  first  obtained  by  Monge,  but  in  a  form  which  rendered 
it  of  little  use.  Strictly  speaking,  however,  this  integral  was 
not  obtained  by  Monge  in  any  form,  but  merely  indicated. 
(See  Monge,  section  on  '*  The  surface  whose  principal  radii 
of  curvature  are  equal,  but  with  contrary  sign" — Section  XX. 
in  Dr.  Liouville's  edition.) 

The  same  integral  was,  according  to  Moigno,  considered 
also  by  Legendre,  and  later  by  Messrs.  Seret  and  Catalan, 
without  obtaining  any  better  results.  Finally,  however,  M. 
Ossian  Bonnet  in  an  article  on  "  The  Employment  of  a  New 
System  of  Variables,"  pubhshed  in  the  fifth  volume  of  the 
Journal  de  Liouville,  i860,  has  shown  that  the  equation  of  the 
required  surface  is  included  under  a  still  more  general  obtain- 


THE  INTERSEPT  PROBLEM.  437 

able  integral  of  comparative  simplicity.  We  present  in  a  note 
Bonnet's  method,  following  the  guidance  of  Moigno,  and  sup- 
plying formulas  and  references,  all  of  which  he  has  omitted. 

370.  It  appears  that  we  can  have  an  infinite  number  of 
surfaces,  all  satisfying  equation  (lo),  but  it  is  evident  that, 
when  the  contour  is  given,  we  are  restricted  to  that  surface 
or  those  surfaces  which  pass  through  every  point  of  the  fixed 
contour,  and  have  at  the  same  time  their  principal  radii  of 
curvature  equal  and  of  contrary  sign.  Or  following  the  anal- 
ogy of  single  integrals,  if  we  suppose  the  general  integral  of 
equation  (lo)  to  have  been  obtained,  we  must  so  determine 
the  arbitrary  functions  which  arise  in  the  integration  as  to 
cause  the  surface  to  pass  through  every  point  of  the  given 
contour.  But  as  we  are  unable  to  present  the  integral  of  (lo) 
in  an  available  form,  we  cannot  give  anything  more  than  this 
general  outline  of  the  treatment  of  the  functions  for  this  prob- 
lem. 

Problem   LVIII. 

371.  Let  V  be  the  portion  of  the  axis  of  z  comprised  between 
the  origin  and  any  tangent  plane  to  a  surface.  Then  it  is  required 
to  determine  the  form  of  the  surface  which  will  minimize  the  ex- 
pression 

the  edges  of  the  surface  being,  as  before,  confined  to  a  fixed  curve. 
It  is  well  known  that  this  intersept  is 
V  ^:^  z  —  px  —  qy^ 
so  that  we  have 


438  CALCULUS  OF  VARIATIONS. 

Here,  then,  we  have 

Vz—  niv'^-^,      Vp=  —  mv'^-'^  X,      Vq  =  —  mv'^-'^j/, 
(^Vpy=  —  mv'^-'^  —  xm{m—  i)v^-'^  v' , 
(J7'^)^=  —  mv'^~^  — ymim  —  i)v'^~^Vj ; 
so  that  the  fundamental  equation 

becomes,  after  dividing  by  'm{m  —  i), 


(2) 


^'""'*;;r^i-  +  ^^'+-^^'}=°'  (3) 


\ 


the  terms  at  the  Hmits  vanishing  as  before,  because  they  involve 
the  values  of  Sz  for  the  contour  only,  which  are  all  zero. 

Now  one  solution  of  (3)  is  evidently  v  =  0;  which  gives 
U  —  o,  and  signifies  that  the  surface  is  a  conic  surface,  having 
its  apex  at  the  origin,  as  all  its  tangent  planes  must  pass 
through  that  point.  But  neglecting  this  supposition,  which 
is  only  a  singular  solution,  and  will  evidently  not  answer  for 
all  supposable  contours,  we  shall  have 

3^'       f-^^'+/^,  =  0.  (4) 


m—  \ 


372.  Equation  (4),  although  in  reality  an  equation  of  the 

second  order,  is  a  partial  differential  equation  of  the  first  order 

yv 
in  V ;  and  being-  written  under  the  form  xv'  -\-  yv,  = ,  is 

easily  integrated  by  the  ordinary  method  for  such  equations 
(see  De  Morgan's  Diff.  and  Integ.  Calc,  p.  203,  where  we  put 

X  =  X,  Y=  y,  z  =  u —  U),  and  Sfives 


/(I). 


(5) 


FORMULM   OF   TRANSFORMATION.  439 

y 

where  /  denotes  any  function  whatever  of  — .  Substituting 
the  value  of  v,  (5)  may  be  written 

xp-\-yq^z-x  '^~=^f  (I).  (6) 

Now  by  putting-  the  second  member  of  this  equation  for  Um 
De  Morgan,  (6)  can  be  integrated  by  the  same  method  as  be- 
fore, and  writing 

m-\-2^\xl  "^^UJ' 
we  obtain 

^=;r^i^g)+;r/'g),  (7) 

where  F  and  f  are  any  functions  whatever. 

373.  Thus  we  have  been  able  in  the  present  case  to  ob- 
tain the  general  integral  of  the  equation  M  —o.  This  integral 
represents  an  infinite  number  of  surfaces  according  to  the 
forms  which  we  assign  to  the  functions  F  and  f.  If,  as 
hitherto,  we  suppose  the  contour  of  the  required  surface  to 
be  some  linear  boundary  fixed  in  space,  we  must  so  deter- 
mine the  forms  of  these  arbitrary  functions  as  to  cause  the 
surface  to  pass  through  every  point  of  this  boundary.  But 
we  shall  consider  the  determination  of  these  functions  more 
fully  hereafter. 


Section  II. 


FORMULM  NECESSAR  Y  FOR    THE    TRANSFORMA  TION  OF   THE 
VARIATION  OF  A    MULTIPIE  INTEGRAL 

374.  We  propose  in  the  present  section  to  present  some 
formulae  which  belong,  strictly  speaking,  to  the  differential 
and  integral  calculus  only,  but  which,  having  been  developed 


440  CALCULUS  OF    VARIATIONS. 

by  M.  Sarrus  for  the  purpose  of  so  transforming  the  variation 
of  a  multiple  integral  as  to  enable  us  to  discuss  more  satisfac- 
torily the  conditions  which  must  hold  at  the  limits,  will  not 
be  found  in  treatises  on  the  ordinary  calculus.  We  begin 
with  a  formula  which  is  generally  known. 

375.  Assume  the  equation   U=^      udx,  where  u  is  some 

function  of  x,  t,  etc.,  /  being  either  a  constant,  or  a  variable 
which  is  to  be  regarded  as  a  constant  in  obtaining  the  inte- 
gral. If  we  change  t  into  /  -|-  ^t,  we  shall,  in  the  most  general 
case  which  can  arise,  have 


6U=u,dx-u,dx,-\-J^  ' 


-— -  St  dx. 

^0       dt 


For  since  /  is  a  constant,  it  is  independent  of  the  general 
values  of  x,  so  that  we  may  vary  it  without  varying  x.  But 
the  values  of  x^  and  x^  are  constants,  and  may,  or  may  not,  be 
independent  of  t.  In  the  former  case  the  first  two  terms  of 
the  last  equation  will  vanish,  but  in  the  latter  they  must  evi- 
dently be  retained.  But  it  is  evidently  immaterial  in  the  last 
equation  whether  we  employ  the  symbol  S  or  d\  so  that  if  we 
regard  dt  as  an  infinitesimal  constant,  we  may  write 

—-        udx  =  dx  A i— V— • 

dt^^o  t/a-o     dt  '       dt  dt 

Or  using  the  sign  of  substitution,  already  explained,  we  have 

-J-  I     udx  —  j      -—  dx  A-  /     u  -— ;  (i^ 

where  it  will  be  seen  that  we  make  the  sign  of  substitution 
mean  also  that  dx,  and  dx,  are  put  for  dx ;  and  this  will  be 
always  the  case  except  it  be  otherwise  indicated.  As  for  dt, 
because  it  is  constant,  dt,  differs  in  no  respect  from  dt,  and  it 


FORMULM   OF   TRANSFORMATION.  44 1 

is  immaterial  whether  we  consider  it  as  controlled  by  the  sign 
of  substitution  or  not. 

376.  Again,  assume  u,  as  before,  to  be  any  function  of 
X,  t,  etc.,  where  t  may  be  any  quantity  independent  of,  or 
dependent  upon,  x.  Then,  in  the  most  general  case,  we  must 
have 

[dii~^^       du  ^_  du  dx 
It!  ~  ~di  ~^~dx~di' 

Hence,  if  we  wish  to  consider  zi  when  x  has  some  particular 
value,  as  x^^  u  will  become  /    u,  and  we  may  write 


/^i         /^i  \du    ,   du  dx  ) 


These  formulas,  which  are  simple  enough,  will  be  found  to  be 
of  the  highest  importance  as  we  proceed. 

377.  Again,  we  evidently  have 

Now  change  u  into    /     udy.    Then  we  have  from  (A\  wricing 
dx  first  to  prevent  confusion, 

But  from  (i)  we  have 

^Jy.    ^^y=Jy.   -dx'^y+L    '^ITx'^ 


442  CALCULUS  OF    VARLATLONS. 

because  when  we  integrate  with  respect  to  y,  x  is  regarded  as 
constant,  and  we  may  therefore,  since  y  is  the  independent 
variable,  put  it  in  the  place  of  x,  and  x,  which  is  now  constant, 
in  the  place  of  /.  We  may  evidently  employ  (i)  in  like  man- 
ner for  any  other  variables,  replacing  x  by  that  variable,  which 
is  then  to  be  regarded  as  independent,  and  t  by  that  which  is 
then  to  be  regarded  as  constant  in  the  integration.  Hence,  by 
substitution,  we  have 

rw/-^  dy  ^pd.  /\  fi = /"  r\  dy.    (B) 

t/^o        t/z/o    dx  *^^o         ly^      dx       '^0  ^2/0        -^  ^    ^ 

Now  in  this  equation  change  u  into  ut.     Then,  observing  that 

the  integral  signs  in  any  term,  having  been  separated  merely 

to  better  indicate  the  distance  to  which  each  extends,  may  be 

again  brought  together,  as  may  also  the  differentials,  we  shall 

,  .  d      ,       udt   ,   du  , 

have,  since  ——ut^^ r  -7—  A 

dx  dx       dx 

I      I     71 —- dy  dx  ^=^  —  I      /      —-tdydx-\-         /     ut  dy 

^Xo    tJyo         dx  *^^o    t/2/0      dx  '^0    ^Vo  -^ 

-^rn\u%d..  (3) 

^^0  /2/0       dx  ^^' 

This  formula  would  evidently  enable  us  to  transform  such  a 
term  as  /       /      V^dp  dy  dx. 

ti,  we  shall  have 

But  from  (2),  putting  x  for  t  and  y  for  ;i',  we  have 
dx'     ^^~'       [d^^d^d^]' 


FORMULAE   OF   TRANSFORMATION.  443 

Therefore  we  have,  by  substituting  in  (C), 

«/^o        /       {dx    ^  dy  dx  \       '^o    / 

Now  if  in  this  equation  we,  as  before,  change  u  into  ut,  we 
shall  obtain 

«^^o   /        dx  ^^0    /      dx  '^0   ' 

-p/\f!L±a.-rr^-^t^^d..  (4) 

^^0    /        dy  dx  ^^0   f      dy     dx  ^^^ 

379.  We  have  also  the  equation 

f-^dy^/\.  (D) 

^Vo    dy  ly^ 

Hence 

/       /      -—dydx=  /     dx  I      -—dy=        dx       u\ 

and  changing  ?/  into  ut,  we  have 

P  r\^dyd.=  -rr^^tdyd.+p /\td..     (5) 

380.  Again,  from  (D),  we  have 

rx:'p"i"C-        <^' 

Whence,  changing,  as  usual,  u  into  ut,  we  obtain 

/T"i*=-/"rf'*+/T»   (^ 


444  CALCULUS   OF  VARIATIONS. 

We  need  hardly  say  that  formulae  (2),  (4)  and  (6)  will  often  be 
replaced  by  others,  in  which  the  suffixes  of  those  quantities 
which  are  written  above  the  sign  of  substitution  only  will  be 
changed  from  i  to  o,  everything  else  remaining  unaltered  ; 
and  observing  this  fact,  equations  (3),  (4),  (5)  and  (6)  are  suffi- 
cient for  the  transformation  of  the  variation  of  a  double  inte- 
gral. 

381.  It  will,  however,  be  convenient  to  deduce  also  in  this 
place  the  formulae  necessary  for  the  transformation  of  the 
variation  of  a  triple  integral.  But  the  reader  is  advised  to 
omit  these  formulae  for  the  present,  returning  to  them  when 
they  are  required. 

382.  If  in  equation  (B)  we  change  u  into  J^  udz,  and  sep- 
arate the  integrals  as  before,  we  shall  have 

/     dx  /     dy-r  L    '^dz  = 

—         dx       -T-        udz  -\- L     /     dy        udz. 

But  from  (i)  we  have 

d    r^^      ,         r'^  du  ^     ,     /^i    dz 
-r  j     udz  =         -J-  dz  -\-  I     Ti  -J-- 
dx*^^o  ^^0  dx  '^0      dx 


Whence,  by  substituting  and  reuniting  the  integral  signs,  we 
have 

^-T-dzdydx=  —  I     7,     /    u-^dydx 

tJxo     e/?/o    *Jzq     dx  ^^0    ^y^     '^0         dx 

—   /      /    -J^        udzdxA-  L     L.    ,L    udzdy. 


FORMULM   OF   TRANSFORMATION.  445 

Now  changing  u  into  ut,  we  shall  obtain 

/      /      /     u-j-  dzdydx  = 

t/^o    t/</o    t/ZQ         dx  "^ 


-/"  /"#  /'«^  d.  d.-rrrut  ^  ^^  ^..  (7) 

383.  Now  it  is 'evident  that  equation  (B)  will  hold  for  any 
two  independent  variables  involved  in  u.  Let  us  therefore 
put  J  for  X,  and  z  for  j/.     Then  we  shall  have 

/      /     —rdzdy=—         /    u^dy-\-         /     u  dz, 
^Vo  '-^2o    dy         -^  ^2/0    '^0      dy    -^    ^  '^o  ^^o 

Therefore,  since  the  integrals  are  all  definite,  we  have 
III     —r-  dz  dy  dx  = 

t/a-o    t/?/o    t/zo     dy 

—  /      /      /    u-r-dydX*-\-  II     udzdx. 
Now  change  u  into  ^/Z,  and  we  have 

III     ti' ~r  dz  dy  dx  —  —  /      /    ^^tdzdydx 

Oxo    t/i/o    t/zo         ^  -^  t/a-o    t/2/0    t/zo     ^  "^ 

+  /     /     /    utdzdx-l      /     /    ut'^dydx.  (8) 

384.  Again  we  have 

/     -—  dz—       u, 

Jz,     dz  '^* 


44^  CALCULUS  OF  VARLATIONS. 

Therefore,  because  the  integrals  are  definite,  we  may  write 
/      /      /     -—-  dz  dy  dx  =^  /  udy  dx, 

tyxo    t/yo    v'zo    d^  -^  t/xo    t/i/o     /«o  "^ 

If  now  we  change  u  into  ut,  we  shall  obtain 

/      /      I     u —-  dz  dy  dx  =z 

Uxo   t/^o   *^zo       clz 

—  /       /      /     —-tdzdydxA-  /      /    ut  dy  dx.  (q\ 

t/xa    t/t/o    ^Zo    dz  '^^o    ^y^     '-^o  "^ 

The  preceding  formulse  will  be  found  sufficient  for  the 
purposes  of  this  work,  although  the  notion  of  variations 
adopted  by  M.  Sarrus,  and  the  generality  of  the  integrals 
which  he  proposes  to  consider,  have  caused  him  to  develop  to 
a  much  greater  extent  than  we  have  done  this  department  of 
the  calculus,  which  might  be  termed  the  calculus  of  substitu- 
tion. Although  we  shall  subjoin  without  demonstration  a  few 
more  formulas  when  we  com.e  to  explain  Sarrus's  notion  of 
variations,  the  reader  who  wishes  to  find  in  a  neat  and  com- 
pact form  the  various  formulae  which  may  present  themselves 
in  this  calculus  of  substitution  is  referred  to  the  Calcul  des 
Variations^  by  Moigno  and  Lindelof,  Legons  I.  and  II. 


PROBLEMS  LVIL  AND  LVIII.  RESUMED.  447 


Section  III. 

MAXIMA   AND  MINIMA    OF  DOUBLE  INTEGRALS  WITH 
VARIABLE  LIMITS 

Problem  LIX. 

385.  In  either  of  the  preceding  problems,  i7istead  of  supposing 
the  contour  to  be  a  fixed  boundary  in  space,  let  it  be  required 
merely  that  its  projection  upon  the  plane  of  xy — tJiat  is,  the  pro- 
jected contour — shall  be  some  fixed  and  closed  boundary.  In  other 
words,  let  it  be  required  that  the  contour  shall  always  touch  certain 
cylindrical  zvalls  of  a  given  form. 

The  terms  cylindrical  and  conic  must  in  this  chapter  be 
understood  in  their  most  general  sense ;  the  first  denoting-  a 
surface  generated  by  the  movement  of  a  right  line  which  re- 
mains always  parallel  to  a  given  line,  and  the  second  that 
generated  by  the  movement  of  a  right  Une  which  always 
passes  through  a  given  point.  Here  the  walls  are  generated 
by  the  movement  along  the  projected  contour  of  a  right  Hne 
which  remains  always  parallel  to  the  axis  of  z. 

It  will  readily  appear  that  in  this  case  the  limiting  values 
of  x  and  y  are  still  fixed,  because  they  belong  only  to  the  pro- 
jected contour,  which  is  fixed,  but  that  z  along  the  contour — 
that  is,  along  the  limiting  walls— is  susceptible  of  variation,  so 
that  Sz  at  the  limits  is  no  longer  necessarily  zero. 

386.  We  are  then  led  by  the  preceding  problem  to  exam- 

ine  what  will  be  the  form  of  SU,  where  U=J^^  J^^    Vdydx, 

V  being  any  function  of  x,  y,  z,  p  and  q,  the  limiting  values  of 
X  only  being  fixed.     We  have  already  seen  that 

SU=r'X'\V,6z-^  V^dpJ^V^dq)dydx',  (l) 


448  CALCULUS  OF  VARIATIONS. 

and  as  that  value  of  (^^was  obtained  under  no  other  restric- 
tion than  that  the  limiting  values  of  x  and  y  should  be  re- 
garded as  incapable  of  variation,  it  must  hold  in  the  case 
which  we  are  considering. 

If  now,  in  equation  (3),  Art.  377,  we  write  u  =  Vp,  t  =  dz, 

dt       ddz       ^  1    11  1 

—  =  -7-  =  Spy  we  shall  have 

S.^S1'  V,Sp dy  dx  =  -/;■£'  S ^'  '^y^^ 

Also  if,  in  equation  (5),  Art.  379,  we  write  u  =  Vq,  t  =  6z, 
t^  —  6q,  we  shall  obtain 

X.  y,.    V^Sqdydx^ 

fxi    f<J,dV„  /'■'■   /"' 

-io  I  -^'^^y^^+l.  L  ^'^''^^-        (3) 

Now  since  the  first  term  in  the  second  member  of  (i)  is  not 
susceptible  of  any  transformation,  combining  these  results,  ob- 
serving that  the  substitution  of  a  quantity  in  the  sum,  differ- 
ence, product  or  quotient  of  two  or  more  functions  is  the  same 
as  if  we  substituted  in  each  function  separately  and  vice  versa, 
we  may  write 

^u=£C  \  V,-  v,%  \  6,dx+/;;fy,s.dy 


'xo   /2/0     [     ^         ^  dx  )  '^0  ^2/0 

which  must,  of  course,  vanish  if  U  is  to  be  a  maximum  or  a 
minimum. 


PROBLEMS  LVII.  AND  LVIII.  RESUMED.  449 

387.  Now  the  projected  contour  need  not  be  a  continuous 
curve,  but  may  be  any  combination  of  right  lines,  curved  lines, 
or  both,  and  we  therefore  speak  of  it  as  a  boundary.  Then 
y^  and  y^  are  the  two  ordinates  of  this  boundary  corresponding 
to  any  given  value  of  x^  and  the  substitution  oi  y^  or  jKo  in  any 
quantity  causes  that  quantity  to  relate  to  the  upper  or  the 
lower  portion  of  this  boundary  only.  To  understand  the 
effect  of  substituting  x^  or  x^  in  any  quantity,  we  observe  that, 
whatever  be  the  form  of  the  projected  contour,  we  must  either 
have  y^  and  y,  zero,  both  when  x  becomes  x^  and  x^,  or  it  must 
consist  in  part  of  a  right  line  perpendicular  to  the  axis  of  x  at 
the  point  x  =  x^  or  x  =  x^,  or  both.  In  other  words,  the 
projected  contour  will  terminate  in  right  lines  whose  equations 
are  x  =  x^  and  x  =  x^.  In  the  latter  case,  then,  the  substitution 
of  x^  or  x^  in  a  quantity  will  cause  it  to  relate  to  these  right 
lines  only,  and  in  the  former  case,  in  which  these  lines  may  be 
regarded  as  becoming  zero,  the  quantity  will  relate  to  the 
points  x^  and  x^  only,  and  will  in  general  vanish. 


388.  Now  writing  k  =  Vq  —  K,^,  the  first  term  of  ^^in 

dx 


dy 
(4),  when  resolved,  becomes 

ll'l'\Szdx-£^'l'"kSzdx,  (5) 

in  which  the  first  integral  is  taken  only  along  the  entire 
intersection  of  the  required  surface  with  that  portion  of  the 
cylindrical  walls  for  which  j^  =:j^,,  and  the  second  along  its 
intersection  with  those  for  which  y  =  y^. 

The  second  term  in  dU,  when  resolved,  becomes 

l''ly,S.dy-l''Jy,S.dy,  (6) 

in  which,  although  the  integration  is  with  respect  to  y,  the 
first  integral  extends  only  along  the  right  line  x  =  x^,  while 


450  CALCULUS  OF  VARLATLONS. 

the  second  extends  only  along  the  line  x  =  x^,  and  either  inte- 
gral will  vanish  of  itself  should  the  projected  contour  not 
terminate  in  the  lines  x  —  x^  and  x  —  x^. 

It  appears,  therefore,  that  the  first  two  terms  of  SU'va  (4) 
involve  only  values  of  ^z  for  the  edge  or  contour  of  the 
required  surface ;  and  also  that  all  these  values  are  included. 
Now  when,  as  in  the  present  case,  we  require  that  the  surface 
to  be  varied  shall  be  comprised  within  certain  cyhndrical 
walls,  the  walls  become  the  limit  of  the  required  surface,  just 
as  do  two  fixed  lines  x  ^:^  x^  and  x  •=  x^,  or  two  fixed  planes 
with  the  same  equations  in  an  analogous  problem  for  curves; 
so  that  the  terms  just  considered  in  SU^  although  still  affected 
by  the  integral  sign,  as  indeed  they  ought  to  be  in  order  that 
they  may  ^um  up  the  variations  of  z  for  the  entire  contour, 
must  be  in  the  variation  of  a  surface  what  the  terms  at  the 
limits  are  in  the  variation  of  a  curve. 

389,  Let  us  now  examine  what  conditions  must  hold  when 
C/is  to  be  a  maximum  or  a  minimum. 

Since  SU  must  now  vanish,  if  we  denote  by  L  the  aggre- 
gate of  the  limiting  terms,  (4)  may  be  written 

wherei/=F, -(Fpy-Cr,),. 

Now  since  the  double  integral  extends  throughout  the  en- 
tire surface,  it  will  appear,  as  in  the  case  of  single  integrals,  that 
we  cannot,  without  in  some  manner  restricting  the  value  of 
Sz^  make  this  general  integral  depend  in  any  manner  upon 
terms  which  refer  solely  to  the  limits,  even  when  those  terms 
are  themselves  under  an  integral  sign.  Therefore  the  terms 
in  (7),  being  completely  independent,  must  be  equated  severally 
to  zero,  so  that  we  shall  obtain,  as  before, 

M=.  V,-{V^y-{V^\  =  o.  (8) 


PROBLEMS  LVII.  AND   LVIIL  RESUMED.  451 

We  shall  also  have,  writing  the  value  of  L  from  (5)  and  (6), 
L=  /      /    kdzdx  —         I    kdzdx 

+ /    X  ^P^^dy-I    l^    V, fc dy  =  o.  (9) 

Now  we  are  evidently  at  liberty  to  vary  z  for  any  portion 
of  the  contour  we  please,  leaving  it  unvaried  for  the  remain- 
der; and  the  four  terms  in  (9)  are  therefore  also  completely 
independent ;  so  that  these  terms  must  also  be  equated  sever- 
ally to  zero,  giving  us  the  equations 


/  k^zdx  —  o,  J     I    kdzdx  =0, 


(10) 


where  the  first  two  integrals  extend  respectively  along  the 
two  portions  of  the  contour  whose  equations  2irQ  y—y^  and 
y  —  y^'^  while  the  last  two,  if  they  exist  at  all,  extend  along 
the  right  lines  whose  equations  are  x  ^  x^  and  x  =  x^. 

But  Sz  along  any  one  of  these  four  portions  of  the  contour 
is  entirely  in  our  power,  while  its  coefficient  is  not.  It  will 
therefore  appear  that  we  can  only  make  the  integral  certainly 
vanish  by  supposing  the  coefficient  of  dz  to  vanish  through- 
out the  whole  range  of  the  integration.  We  must  therefore 
have 

l''"k  =  o,         p^°k  =  o,         /""Vp^o,       l'''Vp  =  o\      (11) 

where  the  substitutions  merely  indicate  to  what  part  of  the 
contour  the  condition  belongs. 

390.  Let  us  now  consider  what  equations  (11)  imply. 
The  first  two  equations  merely  show  that  k  must  vanish 


452  CALCULUS  OF  VARIATIONS. 

along  every  portion  of  the  contour  for  which  ;tr  is  a  variable, 
while  the  last  two  show  that  /^  must  vanish  along  both  por- 
tions of  the  contour  whose  abscissas  are  the  constants  x^  and 
x^,  if  such  portions  exist.  Now  restoring  the  value  of  k^  and 
clearing  fractions,  we  have  along  the  first  two  portions  of  the 
contour 

Vq^dx  —  Vpdj/  =  0.  (12) 

Moreover,  this  equation  holds  also  along  the  other  two  por- 
tions of  the  contour,  when  such  portions  exist,  and  they  furnish 
no  other  condition.  For  along  either  of  these  portions  dx 
vanishes,  while  dj/,  being  taken  along  the  right  line  x  =  x^  or 
X  =  x^,  does  not  vanish,  so  that  it  is  easy  to  see  that  the  appli- 
cation of  (12)  to  either  of  these  portions  would  lead  necessarily 
to  F^  =  o. 

It  appears,  then,  that  equation  (12)  must  hold  for  the  entire 
contour,  and  that  there  are  no  other  equations,  although  we 
have  already  seen  that  this  equation  may  represent  more  than 
one  condition. 

391.  But  before  entering  upon  any  further  discussion,  let 
us  apply  the  results  which  we  have  obtained  to  Probs.  LVII. 
and  LVIIL,  beginning  with  the  former.     Here  (12)  gives  at 

once 

qdx  —pdy  —  o,  (13) 

an  equation  which  indicates  that  the  required  surface  must  at 
every  point  of  its  contour  meet  at  right  angles  the  limiting 
cylindrical  walls.  Now,  theoretically  speaking,  if  we  could 
obtain  the  general  integral  of  equation  (10),  Prob.  LVIL,  in- 
volving two  arbitrary  functions,  these  functions  might  be 
determined  so  as  to  satisfy  two  conditions  at  the  limits.  But 
if  the  limiting  walls  should  consist  of  a  number  of  sides,  curvi- 
linear or  rectilinear,  it  is  evident  that  the  application  of  (13) 
to  each  of  these  sides  might  involve  as  many  distinct  condi- 
tions as  there  are  sides ;  so  that  we  would  expect  in  general 


PROBLEMS  LVIL  AND  LVIII.  RESUMED.  453 

to  find  it  impossible  to  form  a  surface  which  would  satisfy 
the  fundamental  equation  (lo),  and  be  also  at  right  angles  to 
more  than  two  sides  of  a  limiting  wall ;  although  it  might 
happen  that  the  surface  which  would  satisfy  two  ot  these 
conditions  would  satisfy  the  others  also. 

If,  as  is  usual,  we  suppose  the  projected  contour  to  be 
some  closed  curve,  the  limiting  wall  has  but  two  parts,  those 
for  which  y  —  y^  and  y  —y^,  and  all  the  terms  in  dU  affected 
by  the  substitution  of  x^  or  x^  disappear.  In  this  case,  there- 
fore, we  v/ould  infer  that  all  the  conditions  of  the  question 
could  be  satisfied. 

392.  Let  us  now  turn  to  Prob.  LVIII.  Here  we  find  that 
equation  (12)  will  give  the  condition 


p(m 


^^{ydx  —  xdy)^o.  (14) 


Now  as  the  second  factor  of  this  equation  relates  solely  to  the 

projected  contour.,  it  can  become  zero  only  when  the  portion 

of  this  contour  along  which  it  vanishes  is  a  right  line  passing 

through  the  origin.     But  along  any  portion  of  the  contour 

whose  projection  is  not  such  a  right  line  we  must  have  v^o 

But  along  any  portion  of  the  projected  contour  which  is  not  2- 

y 
right  line  passing  through  the  origin,  —     must     be     variable. 

Hence,  in  order  to  satisfy  equation  (5),  Art.  372,/ f-j  and  con- 
sequently i^(-)  must  vanish  throughout   the   entire  surface, 

and  equation  (7),  Art.  372,  will  become  z  —  x/'(-j,  which  is  the 

general  equation  of  a  conic  surface,  having  its  summit  at  the 
origin.  See  De  Morgan's  Diff.  and  Intg.  Cal.,  p.  400,  where 
m,  n  and  /  are  to  be  made  zero. 

Now  this  conic  surface  must  meet  every  element  of  the 
cylindrical  wall,  and  the  function  f  must  be  such  as  to  enable 


454  CALCULUS  OF    VARIATIONS. 

the  surface  to  satisfy  this  condition.  But  it  must  appear  upon 
reflection  that  this  condition  will  not  enable  us  to  determine 
the  form  of/',  since  it  is  evident  that  the  conic  surface  might 
be  of  various  characters  and. still  touch  every  element  of  the 
cylindrical  walls. 

Problem  LX. 

393.  Suppose  we  next  demand  that  in  Prohs.  LVIL  and 
L  VIIL  the  contour  of  the  required  surface  shall  always  rest  upon 
one  or  more  given  surfaces. 

This  case  is  evidently  analogous  to  that  in  which  we  are 
to  connect  two  given  curves  by  a  curve  having  a  certain  maxi- 
mum or  minimum  property,  and  we  can  in  a  similar  manner 
pass  from  the  primitive  to  the  derived  surface  by  first  ascrib- 
ing such  variations  to  z,  p  and  q  as  will  give  us  a  derived 
surface  of  any  required  form,  and  then  so  altering  the  dimen- 
sions of  this  surface  as  to  cause  it  to  intercept  the  bounding 
surface  or  surfaces. 

This  change  of  dimension  will  involve  an  alteration  in  the 
form  of  the  projected  contour,  and  to  consider  this  contour  in 
the  most  general  manner,  we  shall,  as  before,  suppose  that  it 
terminates  in  the  lines  x  =^  x^  and  x  =  x^,  as  we  can  then  easily 
make  the  formulas  thus  obtained  applicable  to  any  other  case 
by  reducing  one  or  both  these  lines  to  points.  We  shall, 
moreover,  for  convenience,  denote  the  four  portions  of  either 
contour  corresponding  respectively  to  jj/  =  y^,  y  =  y^,  x  =^  x^ 
and  X  —  x^  by  the  terms  lozver,  upper,  left  and  right. 

But  in  changing  the  form  of  the  projected  contour  we 
need  not  vary  the  general  values  of  x  or  y,  but  merely  those 
of  J^o>  Ji>  ^0  ai^d  jTj.  For  we  must  remember  that  this  contour 
encloses  a  certain  plane  surface,  and  that  the  general  values 
of  X  and  y,  as  used  in  the  double  integral,  must  include  the 
co-ordinates  of  every  point  of  this  plane  surface ;  so  that  for 
every  value  of  x  there  are  an  infinity  of  values  for 7.    If,  there- 


PROBLEMS  LVII.   AND  LVIII.    RESUMED.  455 

fore,  we  regard  x  and  y  as  varying  throughout  the  integral, 
as  we  evidently  may,  we  in  effect  suppose  the  points  of  this 
plane  to  change  their  positions.  But  this  is  manifestly  use- 
less, it  being  sufficient  to  add  to  the  ordinates  }\  and  )\^  and 
to  the  abscissas  x\  and  x^,  infinitesimal  increments  Dy^  Dy^,  Dx^ 
and  Dx^,  these  quantities  being  independent  of  either  sign, 
and  representing  the  same  increments  as  would  be  denoted 
by  Sy^  and  dx^  and  dx^,  if  we  had  been  obliged  to  vary  in  the 
same  manner  the  form  of  the  projected  contour  in  the  case  of 
a  single  integral,  the  limits  being  also  variable. 

394.  Let  us  now  consider  in  detail  the  mode  of  obtaining 
SUy  where  U  =J^    J     Vdydx,  V  being  any  function  of  x,  y, 

z,  p  and  q,  the  limiting  values  of  x  and  y  being  also  subject  to 
variation. 

First,  varying  z^p  and  q  only,  we  have,  to  the  second  order. 


r^  r\  V,6z+  VpSp+  V^Sqldydx 

+  2V^qSpdq  +  Vq^Sq^\dydx.  (l) 

This  gives  to  the  second  order  the  change  which  6^  will  under- 
go when  we  pass  from  any  primitive  to  any  derived  surface, 
the  form  of  the  projected  contour  or  of  the  bounding  walls 
remaining  unaltered. 

In  the  second  place,  let  us  consider  the  change  which  [/ 
will  undergo  when  we  alter  the  dimensions  of  this  derived 
surface  in  any  infinitesimal  manner  we  please,  supposing 
x^  and  jTj  to  remain  unchanged.     Since  U  consists  of  the  sum 

of  the  elements  dx  I     Vdy,  in  which  the  integration  is  entirely 

independent  of  x^  that  quantity  being  regarded  merely  as  a 


45^  CALCULUS  OF  VARIATLONS. 

constant,  if  it  enter  Fat  all,  the  change  sought  will  evidently  be 
the  sum  of  the  additional  changes  which  each  element  will 
undergo  if,  after  having  varied  z,  p  and  q  only,  we  also  vary 
/o  and  jKi  by  adding  any  increment  or  decrement,  Dy^  and  Dy^, 
Proceeding,  then,  with  one  of  these  elements  as  if  it  were 
the  only  integral  in  question,  we  shall  obtain,  in  addition  to 
the  terms  arising  from  the  variation  of  ^,  /  and  q  only,  which 
are  already  included  in  (i),  the  terms 

C 1  ^'^y  + 1  ^'  ^y^ +^vDy\,  (2) 


where 
and 


K^Vy^V,q+V^s+V^t  (3) 

6V=  V,Sz+  Vpdpi-V^Sq,  (4) 

Hence,  summing  the  changes  in  all  the  elements,  we  have 


VDy-^r\v,D/  +  6VDy  \  dx,  (5) 


which  gives  the  change  sought,  and  must  be  added  to  (i). 

395.  We  must  now  consider,  in  the  third  place,  what 
change  U  will  undergo  when  we  make  any  infinitesimal 
changes  in  the  values  of  x^  and  ;r,. 

It  is  easy  to  see  that  we  can,  if  we  choose,  pass  from  the 
primitive  to  the  derived  surface  by  first  making  the  necessary 
changes  in  y^  and  y^,  or  in  the  form  of  the  upper  or  lower  por- 
tions of  the  projected  contour^  x^  and  x^  remaining  fixed,  as 
also  the  form  of  the  surface ;  and  then  varying  the  surface 
under  the  supposition  that  z,  p  and  q  vary,  and  also  that  x^  and 
x^  become  respectively  x^  -f  Dx^  and  x^  -\-  Dx,,  the  new  limit- 
ing values  of  y,  which  are  y^  -\-  Dy^  and  y^  -\-  Dy^,  remaining 
fixed.     Now  the  portion  of  c^C/ which  will  arise  from  varying 


PROBLEMS  LVII.  AND  LVIII.  RESUMED,  457 

this  surface,  supposing  x^  and  x^  to  remain  fixed,  is,  as  we  have 
seen,  found  by  taking  the  sum  of  equations  (i)  and  (5),  so  that 
we  have  now  to  determine  the  portion  which  will  arise  merely 
from  changing  x^  and  x^  into  x^  +  Dx^  and  x^  +  Dx^,  and  this 
added  to  (i)  and  (5)  vvill  evidently  give  the  complete  variation 
of  U  to  the  second  order. 

396.  Now  by  the  changes  in  the  limiting  values  of  j  alone, 
U  will  become 

P^\   f*V\-\-'Dyx  nx-i 

I  ^      Vdy  dx     or      /     vdx.  (6) 

where 

V  =fjydy^ly  VDy  +  etc.\=  v^+l^  VBy  +  etc.}.     (7) 

Now  since  v  does  not  contain  the  limiting  values  of  x,  either 
explicitly  or  implicitly,  any  element  vdx  will  be  independent 
of  any  changes  in  these  limiting  values,  and  therefore,  although 
2^  is  a  definite  integral,  we  may  employ  the  same  reasoning  as 

though  it  were  not,  and  say  that  the  change  in  J^    vdx,  due 

to  the  variation  of  the  limits  x^  and  x^,  must  be 


1: 1  ■' 


Dx-\^-  v'  Dx"  +  dvDx  \  .  (8) 


Let  us  now  approximate  in  (8)  as  far  as  the  terms  of  the 
seconS  order.     We  have 

=  /"'  f'  VdyDx^  r-  l^'VDyDx,  (9) 

where  we  mu^t  remember  that  the  last  term  represents  four 
terms  involving  merely  the  values  of  V,  Dy  and  Dx  at  what 


458  CALCULUS  OF  VARLATIONS. 

we  may  term  the  four  corners  of  the  surface,  although  Dx^ 
and  Dx^  are  infinitesimal  and  independent  constants. 

Now  in  reducing  the  second  and  third  terms  in  (8)  it  will 
evidently  be  sufficient  to  regard  v  as  merely  equal  to  v"^.  Then, 
by  equation  (i),  Art.  375,  we  have 


and  therefore,  to  the  second  order, 

i^^\  dv  /^i  I    py^dV  /^i  /y^i     dv       „     ,    X 

/    l.'^D^-^^         1/    ^dyDx'  +  l     I    ~Vi-Dx\    (10) 
^^0  2  dx  '^0    2  ^^0    dx    -^  '  '-^0   '^0    2      dx 

^=V'=V,+  V,p  +  Vpr  +  Vqs,  (II) 

where  accents  as  usual  denote  total  differentials.     We  shall 
have  also,  to  the  second  order, 

r^dvDx  =  I'^'S  r^VdyDx  =  r^  r'dVdyDx,        (12) 

where  SV  has  the  value  given  in  (4).     Hence,  adding  equa- 
tions (9),  (10)  and  (12),  we  obtain,  for  the  last  portion  of  ^U, 

'^C-2  V%Dx'-^XyVdyDx  }  .  ■    (13) 

397,  Now  by  adding  (i),  (5)  and  (13),  and  then  substitut- 
ing the  values  of  V,  V^  and  ^Ffrom  (3),  (4)  and  (11),  we  shall 
have  the  complete  variation  of  U  to  the  second  order ;  and  if 
U  is  to  be  a  maximum  or  a  minimum,  the  terms  of  the  first 
order  must  vanish,  while  those  of  the  second  must  become  in- 
variably negative  for  a  maximum  and  positive  for  a  minimum. 


PROBLEMS  LVII.  AND  LVIIL  RESUMED.  459 

As  will  be  naturally  surmised  from  their  complicated  na- 
ture, the  determination  of  the  sign  of  the  terms  of  the  second 
order  transcends  our  present  knowledge  of  variations,  even 
when  the  form  of  V  is  known ;  and  we  shall  therefore  in  future 
consider  only  those  terms  in  dt/ which  are  of  the  first  order. 

Collecting  these  terms,  we  have 

+X'X' 5  ^'^^  +  ^^^^  +  V^6q\dydx  =  O.  (14) 

iVow  transforming  the  double  integral  as  in  equation  (4)  of 
the  preceding  problem,  we  shall  have,  finally, 

+  r^  r^  VDy  dx-\-  T'  T"  VDx  dy 

398.  Now  it  will  appear,  as  in  the  preceding  problem, 
that  because  the  part  of  (^ Sunder  the  sign  of  double  integra- 
tion cannot  depend  upon  terms  which  relate  to  the  limits 
only,  these  two  parts  must  be  independent,  and  that  L  and 
M  must  severally  vanish.  Therefore  we  see  that  here,  as 
in  single  integrals,  the  differential  equation  from  which  the 
general  solution  must  be  obtained  will  be  the  same  whatever 
may  be  the  particular  conditions  which  may  be  imposed  at 
the  limits. 

Let  us  then  examine  the  equation  L  —o.     It  is  easy  to  see 


4^0  CALCULUS  OF  VARIATIONS, 

that  if  we  can  regard  the  quantities  dz^  Dy  and  Dx  at  all  the 
limits  as  independent,  the  four  terms  in  L  will  be  also  inde- 
pendent, and  we  shall  be  obliged  to  equate  them  severally  to 
zero.     Hence,  using  k  as  in  the  last  problem,  we  must  have 

/      /    ^kdzdx  =  o,  /      /    'Vridsdy  =  o, 

\       (i6) 
,  .-  ri'-VDydx  =  o,  r-  r-VDxdy  =  o, 

t/xo     /yo  Ixq    Ugo  -^  J 

in  which  the  first  two  equations  give  the  same  conditions  as  in 
that  problem. 

Now  in  the  third  equation  we  must  remember  that  Dy  is 
perfectly  in  our  power  for  every  point  of  the  upper  and  lower 
portion  of  the  projected  contour,  and  is  in  fact  what  might  be 
termed  ^y^  if  we  had  not  agreed  to  suppose  x  and  y  incapable 
of  receiving  any  variation ;  so  that  this  integral  will  not  cer- 

tainly  vanish  unless  we  have/     F=  c 

In  treating  the  fourth  equation,  we  must  remember  that 
Dx^  and  Dx^  do  not,  hke  Dy^  and  Dy^,  denote  an  infinity  of 
quantities,  but  signify  only  one  each,  so  that  they  are  each 

arbitrary  constants,  and  we  must  have/      /     Vdy  —  o,  and  we 

cannot  make  any  further  reduction,  because  the  integral  is 
definite,  and  none  of  the  quantities  involved  are  in  our  power, 

F=  o. 
We  must  then,  in  the  present  case,  have  the  equations 

k^o,         V=o,         Fp  =  o,         £'Vdy  =  o;        (i?) 

the  first  two  equations  holding  along  the  upper  and  lower  con- 
tour, and  the  last  two  along  the  ri^/it  and  left.  Or,  as  in  the 
preceding  problem,  the  condition  Vqdx  —  Vpdy  =  o  must  hold 
for  the  entire  contour ;  while  we  now  add  that  the  condition 


PROBLEMS  LVII.  AND  LVIIL  RESUMED.  46 1 

F  =  o  along  the  entire  contour  will  satisfy  all  the  remaining 
requirements  of  the  limits,  and  will  be  necessary  for  all  but 
the  right  and  left  portions  of  the  contour,  which  might,  per- 
haps, be  satisfied  by  some  other  condition  also. 

399.  But  as  it  is  necessary  in  the  case  of  curves  to  im- 
pose some  manner  of  restriction  upon  the  extremities  in 
order  that  ^may  become  a  maximum  or  a  minimum,  so  in  the 
present  case  it  is  easy  to  see  that  the  required  surface  cannot 
possess  a  maximum  or  a  minimum  property  unless  its  contour 
be  subjected  to  some  sort  of  restriction. 

Now  the  most  general  case  which  will  arise  is  that  of  our 
problem — namely,  where  the  required  surface  is  to  have  its 
contour  upon  t)ne  or  more  given  surfaces — and  this  case  we 
will  now  proceed  to  consider. 

4-0 0«  Let  the  equation  of  any  one  of  the  limiting  surfaces 
be  of  the  form 

dZ  =  PdX^QdY    or     Z  =  /(X,F),  (18) 

and  let  us  first  suppose  it  to  be  touched  by  a  portion  of  the 
upper  contour.  Now  if  we  pass  a  plane  parallel  to  that  of 
yz,  at  any  distance  x  from  that  plane,  the  sections  cut 
from  the  required  and  the  limiting  surface  will  be  two  plane 
curves,  which  meet,  and  the  equation  of  the  curve  cut  from 
the  limiting  surface  is  dZ  =  QdY,  while  that  of  the  other  is 
ds  =  qdy.  Therefore,  so  far  as  these  two  curves  are  con- 
cerned, we  may  regard  y  as  the  independent  variable,  and  x 
as  a  constant,  if  it  appear  at  all  in  their  equations.  Hence, 
when  we  change  y^  into  y^-\-  Dy^,  we  may  employ  precisely  the 
same  reasoning  as  in  Art.  69  ;  so  that,  since  Q  would  replace/"' 
in  that  article,  we  shall,  neglecting  terms  of  the  second  order, 
have,  as  in  equations  (2),  Art.  ^6, 

Sz,=  {Q-q),Dy,, 

and  a  similar  equation  will  hold  for  the  lower  limit. 


462  CALCULUS   OF  VARIATLONS. 

In  like  manner,  for  the  limiting  surface  at  the  right,  by 
passing  a  plane  parallel  to  that  of  xz  at  any  distance  y  from 
that  plane,  we  find 

and  a  similar  equation  for  the  lower  limit. 

Or  to  render  these  equations  more  intelligible,  we  may 
write 

or,  to  the  second  order,  we  shall  have 

# 

ly.  =iy^i^Q-g)Dy+'-{T-  t)  Df-  SqDy\,^ 

y   (20) 
£"&  =£'  I  {P-p)Dx\\^{R-r)Dx'-SpDx 

401.  Now  since  equations  (19)  restrict  the  independence 
of  ^z  and  Dy^  and  dz  and  Dx  at  both  limits  of  y  and  x,  equa- 
tions (17)  will  no  longer  hold  true.  But  from  (15)  we  may 
write 

^  =£?ly''^  VDy)dx  +  l^jj;;\V^6.  +  VDx)dy  =  o;{2,) 
and  eliminating  Sz  by  (19),  we  have 

+  rX''  ^+  Vp{P-p)]Dxdy  =  o.  (22) 

Now  it  is  evident  that  the  quantities  By^,  By^,  Dx^  and  Dx^  are 
entirely  independent  of  one  another,  as  the  fact  that  the  con- 
tour is  to  be  confined  to  certain  surfaces  in  no  way  restricts 


PROBLEMS  LVIL  AND   LVIIL  RESUMED.  4^3 

US  in  varying  the  form  of  the  projected  contour.  Moreover, 
as  before,  Dy  is  completely  in  our  power  for  every  point  of 
the  upper  and  lower  contour,  while,  for  either  limit  of  x,  Dx  is 
an  arbitrary  constant.  Therefore,  by  the  same  reasoning  as  in 
the  former  case,  (22)  must  give  the  equations 

V^k{Q-q)  =  0,         £'\V+V^{P-p)]dy^O;     (23) 

the  first  holding  along  either  the  idpper  or  lozver  portion  of  the 
contour,  and  the  second  along  either  the  right  or  left.  , 
But  >^  =  Vq  —  Vpj/x  ;  and  also 

tl2  =  pdx  +  qdy         and     dZ  =  PdX  +  QdY, 

the  first  being  the  equation  of  the  required,  and  the  second  of 
any  limiting  surface  ;  and  since  along  their  intersection  ;r,  j/,  ^ 
and  X,  V,  Z  are  identical,  we  must  have  along  such  intersec- 
tion 

dy  _       P-p 


pdx  -f-  qdy  =  Pdx  -\-  Qdy, 


dx  Q  —  q 


Substituting  this  value  in  k,  and  then  the  result  in  the  first  of 
equations  (23),  the  conditions  at  the  hmits  finally  become 


V+Vp{P-p)+Vq{Q-q)  =  o,-] 
r'\V+V,{P-p)\dy  =  c 


(24) 


To  discuss  the  terms  of  the  second  order  we  must  employ 
equations  (20)  in  the  place  of  (19),  proceeding  as  before,  and 
setting  aside  all  terms  of  that  order  which  may  arise.  Then 
we  shall  have  the  same  terms  of  the  first  order  as  before, 
while  those  which  we  have  set  aside  must  be  added  to  the 
terms  of  the  second  order  which  we  have  already  exhibited 
in  equations  (i),   (5)  and  (13),  thus  rendering  the  complete 


4^4  CALCULUS  OF  VARIATLONS. 

terms  of  that  order  still  more  complicated,  and  the  deter- 
mination of  their  sign  a  much  more  hopeless  problem  than 
before. 

402.  Now  the  first  of  equations  (24)  must  hold  along  the 
entire  upper  and  lower  contour,  and  may  represent  as  many 
distinct  conditions  as  there  are  limiting  surfaces  touched  by 
these  portions  of  the  contour.  The  second  of  these  equations 
holds  along  the  right  and  left  contour  only,  and  will  be  satis- 
fied if  we  suppose  the  first  to  hold  for  these  portions  of  the 
contour  also,  because  along  these  portions,  being  parallel  to 
the  plane  of  yz,  q  and  Q  are  equal,  so  that  Q  —  q  will  vanish. 

The  first  condition,  then,  is  necessary  for  the  upper  and 
lower,  and  will  satisfy  the  requirements  of  the  limits,  should  it 
hold  throughout  the  entire  contour,  although  the  right  and 
left  portions  may  furnish  some  additional  condition. 

4-03.  Let  us  now  apply  the  foregoing  theory  to  Probs. 
LVII.  and  LVIIL,  beginning  with  the  former. 

Here  it  is  easy  to  see  that  equations  (24)  give  the  condi- 
tions 

'  +  ^  +  ^^  =  °'     ^(i^i^^^--     (^s) 

The  first  equation  denotes  that  the  required  surface  must 
meet  at  right  angles  all  the  limiting  surfaces  which  are 
touched  by  its  upper  and  lower  contour,  and  the  same  condi- 
tion might  also  prevail  along  the  right  and  left  portions, 
although  we  cannot  assert  that  the  second  of  equations  (25) 
might  not  be  satisfied  in  some  other  manner.  In  general, 
however,  the  projected  contour  will  be  a  closed  curve,  in  which 
case  the  right  and  left  portion  reduce  to  points,  causing  the 
second  equation  to  disappear,  and  the  first  to  hold  along  the 
entire  contour. 

As  before,  if  we  could  obtain  the  general  integral  of  equa- 
tion (10),  Prob.  LVIL,  which  would  involve  a  number  of  arbi- 


PROBLEMS  LVII.  AND  LVIII.  RESUMED.  465 

trary  functions,  not  exceeding  two,  it  would  be  necessary  to 
determine  these  functions  in  such  a  manner  as  to  satisfy  equa- 
tions (25). 

404-.  Let  us  now  turn  to  Prob.  LV^III.     Here  equations 
(24)  become 


V  —  mx{P—p)  —  my{Q  —  q)\  =  o^ 
H'v^n  - 1 1 7;  —  mx{P  —  p)\dy  —  Q. 


(26) 


These  equations  will  both  be  satisfied  hy  v  =  o  throughout 
the  entire  contour,  which  supposition  would,  as  before,  lead 
necessarily  to  a  conic  surface.  Neglecting  this  supposition,  we 
have 

V  —  mx{P  —  p)  —  iny{Q  —  q)  —  o,         qj  =  2  —  px  —  qy. 
Whence,  substituting  and  transposing,  we  have 

-  m{Ppx  +  Pqy)  +  {m  —  i)  {px-^qj)  ^  —  z. 
Adding  mz  to  both  members  and  transposing,  we  have 

m{z  —  Ppx  —  Pqy^  =  (m  —  1)  {z  —  px  —  qy). 

Whence 

z  —  px  —  qy  ni 


z  —  Ppx  —  Pqy       fn 


(27) 


which  shows  that  if  at  any  point  of  the  upper  or  lower  con- 
tour tangent  planes  be  drawn,  the  first  to  the  required  and  the 
second  to  the  limiting  surface,  the  portions  of  the  axis  of  z 
comprised  between  the  origin  and  these  planes  respectively 
will  be  to  each  other  ?iS  mis  to  m  —  i. 

406.  Having  now  reached  the  general  discussion  of  the 
problem,  let  us  consider  more  particularly  the  mode  of  deter- 


466  CALCULUS   OF  VARIATIONS. 

mining  the  arbitrary  functions  in  the  various  cases  which  may 
arise. 

First  suppose  the  contour  to  be  a  fixed  boundary,  and  let 
it,  for  example,  be  a  circle  of  radius  a,  having  its  centre  on  the 
axis  of  z,  and  its  plane  parallel  to  that  of  xy  at  the  distance 

c.     Write  —  =  /  and  n  =  — - — .     Then,  from  the  equation  of 
X  I  —  m 

the  contour  and  from  the  general  equation  of  the  surface, 

which  now  becomes 

z  =  x^F{t)  +  x/Xt)  =  x^  F+  xf,  (28) 

we  have 


/  {^+y)  =  ^\       /  =^- =  I,       /  ^  =  c, 

'  '  a  * 


(29) 


Having  solved  the  last  equation  for/',  we  may  then  omit  all 
signs  of  substitution,  because  the  form  of  /'  must  remain  the 
same  for  all  values  of  x  and  y  belonging  to  the  required  sur- 
face.    Hence  we  have 

/r+?  ( ^  _       a:^F      \ 
^~        a        \  i/(T+7^)^(*  (30) 

Now  restoring  the  value  of  /,  and  substituting  for/'  in  (28), 
we  obtain 


As  the  lower  limiting  values  of  y  furnish  the  same  equations 
as  the  upper,  we  have  no  other  condition  by  which  to  deter- 
mine F,  which  may  therefore  be  assumed  arbitrarily. 


PROBLEMS  LVII.  AND  LVIII.  RESUMED.  467 

Next  suppose  two  circular  arcs  situated  as  before,  having 
radii  a  and  a' ,  and  that  the  given  contour  is  to  consist  of  a 
portion  of  the  upper  arc  of  each  circle  joined  by  any  two 
curves  whose  projection^  on  the  plane  of  xy  shall  be  the 
right  lines  x  —  x^  and  x  —  x^.  Then  y.^  and  y^  belong  to  these 
arcs  only,  and  we  obtain,  as  before, 


=/ 


4/(1+0 


V(l+/=)»      '       VI +1' 


(32) 


Solving  these  equations  for/'  and  F,  and  omiting  the  signs  of 
substitution  for  the  sanle  reason  as  before,  we  have 


Substituting  these  values  in  (28),  we  obtain 

z(aa'^  -  a'a^)  =  Qa'^  -  ca^)  Vx'  +  /+  {ca  ~  ca')  V(?+7)'^.(34) 

Thus  we  see  that  the  two  functions  will  be  determined  by 
the  circumstance  that  the  required  surface  is  to  pass  through 
the  two  arcs,  and  we  cannot  impose  any  further  conditions. 
Unless,  therefore,  the  remaining  portions  of  the  fixed  bound- 
ary be  so  assigned  that  they  would  lie  necessarily  upon  this 
surface,  the  conditions  of  the  problem  cannot  be  all  satisfied. 
We  shall,  however,  have  occasion  to  consider  these  functions 
again  presently. 

406,  Let  us  next  suppose  that  the  required  surface  is  to 
connect  two  planes  whose  equations  are 

2  =  ax  -\- dy -{-  c        and     ^  =  a'x  +  d'y  -|-  ^'.  (35) 


468  CALCULUS  OF   VARLATLONS. 

From  equation  (5),  Art.  372,  observing  that 

m±2_^        and     n  =  -^~,  (36) 

we  have  v,  or  the  numerator  of  (27),  equals 


fn  —  I 


^F,  (37) 


while  the  denominator  of  that  equation  must  become  either 
c  or  c' .     Hence  (27)  furnishes  the  conditions 

/Vx  IVo 

{m  +  2)x'^F  =  mc,         /    (^  +  2)x'^F  =  mc',        (38) 

We  have  also,  along  the  upper  contour, 

z  —  {a-[-  bt)x  A-c,         z  =  x^F-\-  xf\  (39) 

Eliminating  z  and  x^F  between  these  and  the  first  of  (38),  we 
obtain 

2c 

''  =  (m-^2){f  -a~bt)  '*  (4^^ 

and  substituting  this  value  in  the  first  of  equations  (38),  we 
have 

{f  -  a  —  btf  =  —  {m-\-  2y-'^c''-^ F\  (41) 

and  in  like  manner  we  find,  along  the  lower  contour, 

{f  -a'  -  b'ty^  =  —  (//^  4-  2)1  -nc'n-\F,  (42) 

fi I 

If  we  solve  (41)  and  (42)  for/'  and  F,  and  put  /  for ,  we 

iz 

shall  obtain 


PROBLEMS  LVII.  AND  LVIII.  RESUMED.  4^9 


t    —  ^/p  _  ^p  ' 


\  (43) 


,      ,     s      .  \a-\-bt  —  a'  —  b't\'^     \ 
F=m(m  +  2Y-  \     \^,,_^,^        f    •   J 

Although  in  (4 1)  and  (42)  f  belonged  respectively  to  the 
upper  and  lower  projected  contour  only,  in  (43),  for  the  rea- 
son already  explained,  it  may  belong  to  any  point  whatever  of 
the  required  surface.  Hence  equation  (28)  becomes,  after  re- 
storing the  value  of  /, 

_  c'^iax  -^  by)  —  cP(a' x  +  b'y) 
^  ~"  c'P  —  c^ 

,      ^      ^   \ax-\-by  —  a'x  —  b'y}'^  .     ^ 

+  m{m  +  2)^-1  I        ^^^-^.^  _  ^^^  -^  [.  (44) 

407.  It  will  be  remembered  that,  in  the  case  of  maximiz- 
ing or  minimizing  any  single  integral  U,  it  is  necessary,  in 
order  to  render  the  method  of  variations  applicable,  that  no 
element  of  U  ox  oi  STJ  shall  become  infinite  within  the  range 
of  the  integration  ;  and  it  will  readily  appear  that  when  ^is  a 
definite  double  integral  the  same  principles  will  apply,  since 
each  element  of  U  is  treated  precisely  as  before.  Now  from 
(37)  we  have,  in  the  present  case, 

U=r'r\'^  dydx  —  fj'f^'  c^^^nrnpyn  dy  dx,        (45) 

fyi    I    2  "^fn 

where  C  = ■ — .     But  nm  = ;  so  that  it  will  appear, 

m  —  \  \  —  m  ^^ 

upon  a  little  reflection,  that  nm  must  be  negative  except  when 
m  lies  between  zero  and  unity.  Hence  when  x  ^  o,  x'^'^  must 
become  infinite  ;  and  it  will  appear  that  to  prevent  v  from  be- 
coming certainly  infinite,  or  at  least  indeterminate,  we  shall 


470  CALCULUS  OF  VARIATIONS, 

be  obliged  to  make  F  vanish  throughout  U^  which  will  give 
z  —  xf\  thus  bringing  back  the  conic  surface,  in  which  we 
must,  as  V  is  zero  when  x  is  zero,  still  reject  all  values  of  m 
which  would  render  v^,  ^m-i  qj.  ^m-2  infinite,  since  the  second 
and  third  of  these  quantities  occur  respectively  as  factors  of 
the  terms  of  the  first  and  second  orders. 

In  this  case,  there  being  but  one  function  to  determine, 
the  first  supposition  in  Art.  405  would  determine  the  surface 
completely,  requiring  a  right  cone  ;  so  that 


In  Art.  392  f  would,  as  we  have  seen,  remain  indeter- 
minate, and  indeed  it  is  easy  to  see  that  we  could  have  no 
finite  minimum  while  the  limiting  values  of  z  remain  variable. 
In  this  case  equation  (27)  is  inapplicable,  since  in  obtaining  it 
we  assumed  that  v  did  not  vanish. 


Problem  LXI. 

4-08.  It  is  required  to  determifie   the  form  of  the  surface 
zvhich  will  maximize  or  minimize  the  expression 

u--£?fy!'  ^/+?  ^yd-  =£?£'  vdy<^'-        (I) 

Here 

and  observing  that/,  —  /  =:  s,  the  equation  M  =  o  will  reduce 
to 

q'r  -2pqs+p''t  =  0.  (3) 

This  equation  may  be  integrated  by  the  method  of  Monge ; 


ADDITIONAL  PROBLEMS.  4/1 

and  adopting  the  notation  of  De  Morgan,  page  719,  we  have, 
putting  z  for  u, 

R  =  q\        ^=-2/^,         T=p\ 

a  =  q^df-{-  2pq  dy  dx  -\-  p^dx^,         V  —  o, 

P 
G  =  q^dp  dy  +  p'dq  dx,  -^  —  —  ^,         dz  ■=  pdx  -f  qdy. 

Now  if  a  vanish,  we  see  from  the  last  equation  that  dz  will 

vanish  also,  and  vice  versa;  and  by  theory  cr  will  also  vanish. 

Substituting  j.i  dx  for  dy.,  the  equation  o"  =  o  gives  qdp  —  pdq  =  o. 

i> 
Whence  we  may  write  -  =  —f{z)  —  —f.    Again,  when  or  =  o, 

we  have 

dy  —  f^dx    or     dy-\-^dx    or     dy  —  fdx-=o,  (4) 

where  we  must  remember  that  /  is  to  be  regarded  as  a  con- 
stant, because  dz  is  zero  or  ^  is  a  constant.     Hence 

y—xf=  F{z)  =  F. 

The  complete  integral  of  (3)  is,  then, 

y  =  xA^)  +  F{^)  =  ^f+P,  (5) 

where  /  and  F  are  any  functions  of  z  whatever. 

409.  Let  us  first  suppose  the  limiting  values  of  x^ya-ud 
z  to  be  fixed,  or  that  the  surface  is  to  pass  through  some  fixed 
boundary,  and  let  us  require,  as  a  particular  case,  that  two 
portions  of  this  boundary  shall  be  given  by  the  equations 

/v+/)=^s  r^=r^,      (6) 

/V+/)==^'^       /%./--$.  (7) 


472  CALCULUS  OF  VARIATIONS. 

Then,  for  the  upper  limit,  we  have 


zx  _  a  a  am 


^,^1.^    i/:+f;     ^-"  +  ^ 


m 

r  I  +  =^,     r  I  +    ^ 

m 


az 

y  = 


Vm'-\-z' 
Therefore,  by  (5),  we  have,  for  the  upper  Umit, 


az        __  ^^^_        I        p^  (g^^ 


^/W^z''       Vm'  +  z' 
Similarly  we  obtain,  for  the  lower  limit  of  j, 

a'z        _       a'm'f 


Vm'''  +  z'        Vm' 


+  F,  (9) 


Solving  for  /  and  F,  remembering  that  the  results  will  no 
longer  refer  to  the  contour  only,  but  will  hold  for  every  point 
of  the  required  surface,  we  shall  obtain 


a  Vm'''  -{-z"  -  a'  Vnf+ 
f=z 


am  ym'^  -\-  z^  —  a'm!  Vm^  -\-  ^ 

„  _  aa\m  —  m-'^z 

am  Vjn'^  -\-  z^  —  a'm'  Vm^  -\-  ^ 

Now  if  the  surface  determined  by  the  substitution  of  these 
values  of /and  i^in  (5)  do  not  necessarily  fulfil  all  the  require- 
ments of  the  problem  regarding  other  portions  of  the  fixed 
boundary,  we  conclude  that  these  conditions  cannot  all  be 
satisfied. 

410.  Next,  suppose  we  give  merely  the  limiting  values  of 
X  and  y^  those  of  z  remaining  variable ;  that  is,  suppose  we 


ADDITIONAL  PROBLEMS.  473 

give  merely  the  form  of  \h^  projected  contour,  or  of  the  cyHn- 
drical  walls.  Then  we  see  from  (2)  that  equation  (12),  Art. 
390,  will  furnish  the  condition 

qdx  —  pdy  —  o ;  (lo) 

which  shows  that  the  required  surface  must  meet  these  walls 

at  right  angles. 

To  discuss  the  form  of  the  functions,  let  us  suppose  the 

wall  to  be  a  right  circular  cylinder,  having  the  axis  of  z  as  its 

xdx 
axis.     Then  along  the  projected  contour  we  have  dy  ^ , 

y 

and  (10)  gives,  by  substitution, 

px-\-qy  =  o.  (II) 

But  by  differentiating  (5)  with  regard  to  x  and  y  respectively, 

we  find 

—  xf  y  xfA-F 

px  =  -      /     ,         qy  =      .    ,       -  =     ;;    '       . 

Hence  (11)  gives  F^o;  and  (5)  becomes y  =  xf,  which  may 
evidently  be  put  under  the  form 

The  function  /~*  will  remain  undetermined  unless  we  as- 
sume some  other  form  for  a  portion  of  the  cylindrical  wall. 
Suppose,  then,  another  portion  to  be  elliptical,  giving 

bx  dx 

-^  ay 

Then  along  it  we  have,  as  before, 

# 

bpx -\- ^qy  =  O'  (13) 


hold 

> 

px-. 

^xff 

-v,=.- 

-ft 

Whc 

;nce 

(13)  gives 

(fl 

-b)ff 

-I  

0, 

474  CALCULUS  OF   VARIATIONS. 

y 

Putting  /  for -,  we  shall  have  from  (12),  which   raust  now 


qy'=^yft  "ty=f,  »^. 


411.  Let  us  next  suppose  the  edges  of  the  surface  are  re- 
quired merely  to  rest  upon  one  or  more  given  surfaces.  Then, 
substituting  from  (2)  in  equations  (24),  Art.  401,  we  find  the 
conditions  at  the  limits  to  be 

Pp-\.Qq^O,  r'-^=^M=^dy=:0.  (14) 

The  reader  can  readily  apply  in  any  particular  case  the  first 
of  these  conditions  to  the  determination  of  the  arbitrary  func- 
tions. 

412.  When  the  limiting  values  of  x  and  y  are  fixed, 
whether  those  of  z  be  subject  to  variation  or  not,  we  find  the 
terms  of  the  second  order  to  be 

Hence,  since  we  suppose  the  denominator  of  (15)  to  be  posi- 
tive, we  may  conclude  that  U  will  become  a  minimum  for  ali 
solutions  which  do  not  give  rise  to  infinite  values  for  any 
element  of  dU]  unless,  indeed,  it  be  possible  to  assign  such 
values  to  Sp  and  Sq  as  will  cause  every  element  of  (15)  to 
vanish. 


ADDITIONAL  PROBLEMS.  475 


Problem  LXIL 


413.  It  is  required  to  determine  the  form  of  the  surface  "juhich 
will  maximize  or  minimize  the  expression 

/      /    \z  —  px  —  gy)  dy  dx  =  I   ^vdydx,  (i) 

while  at  the  same  time  the  vacations  of  p  and  q  are  always  to  be 
so  taken  that  the  expression 

may  always  have  an  assigned  constant  value. 

This  is  evidently  a  problem  of  relative  maxima  and  mini- 
ma, and  we  can  treat  it  by  Euler's  method  precisely  as  in  the 
case  of  single  integrals.  For,  supposing  first  the  limiting  val- 
ues of  x,  y  and  z  to  be  fixed,  the  reasoning  of  Bertrand,  ex- 
plained in  Art.  93,  which  the  reader  is  supposed  to  re-peruse, 
can,  in  the  following  manner,  be  extended  to  this  problem. 

Since  the  terms  at  the  limits  vanish,  we  must  have 


/      /    'Svdydx     or      /      /    ^VSzdydx  =  o. 

e/Xo    e/Z/o  *Ixo    <^2/o 

r^  r^Sv'dy  dx     or      f'  H"  V'dzdydx^  o ; 


(3) 


'Xo    t/yo  vxo    t.fyo 

where 

77  —  7,  _  ^!i  _  fl^  V  —  '?)  '  —  '^^'^ — ^-  (a\ 

"^-^^       dx         d/  ^    ~^"  dx  dy  '  ^4^ 

Now  suppose  the  required  surface  to  have  been  obtamed,  and 
on  it  select  any  two  portions  in  such  a  manner  that  for  every 
point  of  either  portion,  when  that  portion  is  considered  sepa- 
rately, both  V  and  V  may  preserve  an  invariable  sign.    Then 


4/6  CALCULUS  OF  VARIATIONS. 

vary  z  throughout  these  portions  only,  leaving  the  remainder 
of  the  surface  unvaried  in  form.  Also  make  the  sign  of  dz 
invariable  throughout  each,  giving  to  it  in  the  two  portions 
like  signs  when  those  of  V  are  unhke,  and  vice  versa. 

In  this  way,  by  giving  suitable  values  to  Sz,  we  can,  as  in 
Art.  93,  satisfy  the  first  of  equations  (3).  But  the  second  of 
these  equations  may  be  written 

X  X  '^'^y''^  =  X.  Jy,  fVS.dydx,        /=f ;  (5) 

the  variations  of  z  being  taken  as  before  ;  so  that  unless  f  be 
a  constant,  we  can  certainly  effect  that  the  double  integrals 
taken  throughout  the  two  portions  shall  be  numerically  un- 
equal, and  hence  the  second  of  equations  (3)  would  not  be 
satisfied. 

The  remaining  reasoning,  by  which  the  necessity  of  Euler's 
method  is  established,  is  precisely  like  that  of  Art.  93. 

If  the  limiting  values  of  x,  y  and  z  are  also  subject  to  varia- 
tion, the  method  of  Euler  is  still  equally  applicable.  For  sup- 
pose the  required  surface  were  to  be  bounded  by  certain 
cylindrical  walls  or  by  certain  surfaces.  Then,  since  we  are 
not  compelled  to  vary  the  limiting  values  of  x,  y  or  ^,  the  re- 
quired surface  must  evidently  be  of  that  kind  which  will  sat- 
isfy all  the  conditions  of  the  problem  when  the  contour  is  to 
be  fixed,  the  only  question  being  to  determine  the  conditions 
which  must  hold  along  the  contour ;  and  since,  in  double  as  in 
single  integrals,  the  fundamental  equation  obtained  in  discuss- 
ing any  problem  of  absolute  maxima  or  minima  is  the  same 
whatever  be  the  conditions  which  are  to  hold  at  the  limits,  the 
appHcability  of  Euler's  method  is  apparent,  as  in  Art.  96. 

4(4.  We  see,  then,  that  we  are  in  the  present  case  to  dis- 
cuss the  conditions  which  will  maximize  or  minimize  abso- 
lutely the  expression 


ADDITIONAL   PROBLEMS.  477 

^  =  £?J2'^^  -P^-qy^a  •//+?)  dy  dx 


Here 


F,=  i,         Fp=-;r  +  — ^=,         F^  ==__;/  + _^£=  ;  (2) 
SO  that,  writing  ^  z=  —  — ,  the  equation  M  =  o  gi^^es 


^V  -  2J>qs  +/V  =  ^(/'  +  /)».  (3) 

This  equation  is  integrable  by  the  method  of  Monge.  See 
Boole's  Diff.  Eqs,  Chapter  XV.,  or  De  Morgan's  Diff.  and 
Integ.  Calc,  page  719.  Adopting  the  notation  of  the  latter,  we 
may  write 

Zp_  ^'  C_  2/^  7-_  /  jr_r 

yp  -\-q)^ 

^  _  q\dq dy  -  dpdx)  +  2pq dqdx   ,  ^  ^^ 

(7+??  +       ' 

dz  ^=  pdx  -\-qdy. 

Now  the  condition  a:  =  o  renders  dz  zero,  and   also  gives 
dy  —  }xdx  =:.  Q\  so  that  we  may  write 

ft  dx  =  -  ^+/.(^)  ^-y  +/..  (A) 


478  CALCULUS  OF  VARIATIONS. 

Now  substituting  pidx  for  dy  m.  g  and  r,  which  must  also  be- 
come zero  when  a  is  zero,  we  obtain 

Integrating  these  equations,  we  obtain 
_Z__z  =  -b  r^dx^ flz\     --=i=  =  -bx  +fAz\     (B) 

Now  by  squaring  and  adding  equations  (B),  and  substituting 
from  (A)  the  value  of  /  -  dx,  we  shall  obtain  the  integral 
sought. 

The  complete  integral  of  this  equation  is,  therefore, 

L  =  (^  +/(^))'  +  (j/  +  F{z)y  =  {x  +/)'  +  (j.  +  F)\     (4) 

415.  This  equation  is  easily  interpreted.     For  suppose  a 

circle  whose  radius  is  — ;  and  while  keeping  its  plane  always 

parallel  to  that  of  xz^  let  its  centre  move  along  some  curve 
in  space  whose  equations  shall  be 

X^-f.         Y=-F,        Z^z,  (5) 

Then  it  will  readily  appear  that  (4)  represents  the  equation  of 
the  surface  generated  by  the  circumference  of  this  circle  as  it 
moves  along  the  given  curve,  and  that  when  we  shall  in  any 
particular  case  have  determined  the  form  of  the  two  arbitrary 
functions,  /  and  F,  we  shall  know  the  nature  of  the  curved 
directrix  of  this  surface.  When  the  contour  of  the  surface  is 
fixed,  the  functions  must  be  determined  in  accordance  with 
this  condition. 


ADDITIONAL   PROBLEMS,  479 

If  the  bounding  walls  only  are  to  have  a  given  form,  equa- 
tion (12),  Art.  390,  will  give 


(/  ^P"  +  /  -  (^Q¥x  =  {x  y/  -^q"  -  ap)dy,  (6) 

and  /  and  F  must  be  determined  so  as  to  satisfy  this  equation. 
When  the  required  surface  is  to  be  limited  by  one  or  more 
given  surfaces,  the  first  of  equations  (24),  Art.  401,  which  is 
the  only  one  of  importance,  will  become,  by  substituting 
from  (2), 

^/  +  q"  ^^^ 

and  /  and  F  must  then  be  determined  in  accordance  with  this 
condition. 

416.  Of  these  cases  we  will  consider  but  one — that  in 
which  the  required  surface  is  to  be  limited  by  two  planes, 
each  passing  through  the  origin,  and  having  for  their  equa- 
tions 

z  ^  ex  -\-  c^y,         2^=-  c'x  -\-  c^y,  (8) 

In  this  case  (7)  will  give 

l''\cp  +  c,q)  =  O,         /\j>  +  clq)  =  a  (9) 

But  from  (4)  we  obtain 


/=-         -+^ 


?  =  - 


(10) 


(^+/y.+0'+^)^/j 

Hence,  by  the  use  of  (8),  equation  (9)  gives 

/'\^J^cf+c,F)^o,         /%J^c'f+c,'F)=o.      (II) 


48o  CALCULUS  OF  VARIATIONS, 

From  (5)  these  conditions  may  be  written 

Z=cX-^c,Y,        Z=c'X+c/Y.  (12) 

From  these  equations  it  at  once  appears  that  if  there  were  but 
one  limiting  plane,  the  centre  of  the  generating  circle  would 
be  compelled  to  remain  always  in  that  plane,  and  that  in  the 
present  case  the  centre  must  move  along  the  intersection  of 
the  two  limiting  planes.  This  will  give  an  oblique  cylinder 
having  a  circular  base,  the  line  in  which  the  two  planes  inter- 
sect being  its  axis.  We  can,  of  course,  determine  /  and  F  in 
the  usual  way,  thus  obtaining,  from  (11), 

cc^  —  c  c^  cc^  —  c  c^ 

Moreover,  when  in  any  particular  case  we  have  determined 
the  functions  /  and  F^  we  shall  then  be  able  to  determine  also 

the  constant  ^  or  —  — .     For,  as  in  the  case  of  single  integrals, 

we  have  the  condition  that  one  of  the  double  integrals  is  to  re- 
main constant,  and  we  may  suppose  a  definite  value  to  have 
been  assigned  to  it. 

4 17.  In  considering  the  terms  of  the  second  order  the 
same  reasoning  will  hold  as  in  the  case  of  single  integrals. 
For  the  variations  of  ^,/  and  q  are  subject  to  a  certain  restric- 
tion which  we  cannot  explicitly  express,  and  the  method  of 
Euler  will  cause  the  terms  of  the  first  order  to  vanish  whether 
these  variations  are  restricted  or  not.  But  the  variations  are 
still  restricted,  and  when  we  come  to  the  terms  of  the  second 
order  it  is  conceivable  that  even  when  they  do  not  indicate  a 
maximum  or  a  minimum,  the  variations  being  unrestricted, 
they  would  do  so  if  we  could  employ  such  variations  only  as 
would  permit  one  of  the  double  integrals  to  remain  always 
constant,  which,  however,  we  have  no  means  of  doing.  But 
when  these  terms  indicate  an  absolute  maximum  or  minimum 


SURFACE   OF  LOWEST  CENTRE   OF  GRAVITY.  48 1 

— that  is,  for  all  systems  of  variations — there  would  seem  to 
be  no  doubt  as  to  the  existence  of  a  relative  maximum  or  mini- 
mum also. 

In  the  present  problem,  when  the  limiting  values  of  x  and 
y  are  fixed,  the  terms  of  the  second  order  are  the  same  as  in 
equation  (15),  Art.  412,  only  multiplied  by  a.  Hence  we  may 
in  this  case  conclude  that  t/will  be  a  maximum  or  a  minimum 
according  as  a  is  negative  or  positive. 


Problem   LXIII. 

4- (8.  It  is  required  to  determine  the  form  which  a  surface  of 
given  area  whose  edges  are  in  some  manner  confined  must  assume 
in  order  that  the  depth  of  its  centre  of  gravity  may  be  a  maximum. 

The  given  area  is 

and  assuming  the  axis  of  z  vertically  downward,  we  have,  for 
the  depth  of  the  centre  of  gravity, 

which  is  to  be  a  maximum.    Or,  since  A  is  to  be  a  constant,  we 
may  say  that  /    '  f^^z  V  i  -\- p""  -\-  q"  dy  dx  is  to  be  a  maximum, 

while  J^    J     4/1  +/'  -\-q^  dydx  is  to  remain  constant.    Hence, 
employing  Euler's  method,  we  may  write 

^=X?fyy  -  ^)  ^^i+f  +  fdydx  =£;£vdydx.    (I) 


482  CALCULUS  OF    VARLATIONS. 


Here  F^r.:  |/i -[-/  +  ^^ 

Y  ^        (^  -  d)p  p  ^        {z-d)q^ 


(2) 


Hence  the  equation  M  =  o  reduces  to 

i+/  +  ^^-(^-^^)Ki+/)^-2/^^  +  (i+/)^}.        (3) 

This  equation  is  not  integrable ;  but  calling  R  and  R '  the 
principal  radii  of  curvature,  and  estimating  the  signs  properly, 
(3)  may  be  written 


because 


=r  — : —  -y  (4) 

R        R'       {z-a)Vl-\-f  +  q' 


Equation  (4)  shows  that  the  mean  radius  of  curvature  of 
the  required  surface  at  any  point  is  twice  the  normal  ex- 
tended until  it  meets  the  plane  whose  equation  is  ^  =  <^.  The 
same  equation  also  indicates  an  analogy  between  this  surface 
and  the  catenary,  which  gives,  as  we  have  already  seen,  the 
solution  for  a  similar  problem  relative  to  plane  curves.  (See 
Art  282.) 

If  the  contour,  instead  of  passing  through  some  fixed 
curve,  be  confined  to  certain  cylindrical  walls  only,  we  must 
have,  from  equation  (12),  Art.  390,  qdx—pdy^=^o,  showing 
that  the  surface  sought  must  meet  these  walls  at  right  angles. 

When  the  edges  of  the  required  surface  must  be  upon  one 
or  more  given  surfaces,  the  equation  of  any  one  of  which  is 
dZ  =  PdX -\- QdY,  the  first  of  equations  (24),  Art.  401,  will 
give  the  condition  i  -{-  Pp  -\-  Qq  =  o,  showing  that  the  required 
surface  must  be  normal  to  the  limiting  surfaces. 


COVERING  SURFACE   OF  MINIMUM  AREA.  4^3 

Problem  LXIV. 

4- 19.  It  is  required  to  detennirie  the  form  of  the  surface  whose 
area  shall  be  a  minimum,  and  which  shall  cover  a  given  volume  on 
a  horizontal  plane. 

Here,  since  the  given  volume  is  /  /  zdydx,  we  may 
write  at  once 


Here 


and  the  equation  M  =  o  will  give 

(I  +  qy-  2pqs+{l  +f)t,l_  ^  ^ 

V^T+f  +  T?  ^        *  ^^^ 

This  equation,  which  is  not  integrable,  gives,  as  in  the  preced- 
ing problem,  by  a  contrary  estimation  of  signs, 

R  +  J'  =  a-  (^ 

Hence  the  required  surface  must  be  such  that  its  mean  cur- 
vature at  every  point  may  be  constant. 

4-20.  We  already  know  that  it  will  be  necessary  to  the 
existence  of  a  maximum  or  a  minimum  that  the  contour  shall 
either  be  fixed  or  rest  upon  some  surface  or  surfaces,  the  cal- 
culus of  variations  affording  in  the  first  case  no  further  equa- 
tions; and  we  are  unable  to  integrate  (3).     But  when,  in  the 


4^4  CALCULUS  OF  VARIATIONS. 

second  case,  these  limiting  surfaces  are  certain  cylindrical 
walls  normal  to  the  plane  of  xy,  equation  (12),  Art.  390,  gives 

qdx—pdyz=zo^  (5) 

the  meaning  of  which  we  know. 

When,  however,  the  limiting  surfaces  to  which  the  contour 
is  to  be  confined  may  have  any  given  form,  the  first  of  equa- 
tions (24),  Art.  401,  gives 


^  Vi  +/  +  ^^  -  a(i  +Pp+  Qq)  =  o.  (6) 

Suppose,  for  example,  the  limiting  surface  to  be  a  plane  whose 
equation  is  ^  =  ^     Then  (6)  will  give 

(7) 


Hence  the  angle  A  which  the  tangent  plane  to  the  required  sur- 
face at  any  point  of  the  contour  makes  with  the  plane  of  xy 

must  be  a  constant,  since  the  first  member  of  (7)  is or 

sec  A 

cos  A. 

When  h  —  o,  we  must  have  cos  ^  =  o,  and  the  required 
surface  meets  the  plane  of  xy,  and  is  normal  to  it.  The  sur- 
face of  a  hemisphere  of  radius  2a  would  evidently,  in  this  case, 
satisfy  all  the  conditions  of  the  question  so  far  as  the  terms  of 
the  first  order  are  concerned  ;  but  a  satisfactory  investigation 
of  those  of  the  second  order  would  probably  be  impossible. 

When  the  limiting  values  of  x  and  y  are  fixed,  the  terms  of 
the  second  order  may  be  written 

and,  as  in  the  case  of  a  spherical  surface,  the  radius  is  2a,  and 
is  positive,  we  may  conclude  that  if  we  vary  the  form  of  the 
surface  only,  the  circular  base  remaining  unvaried,  the  surface 
will  be  a  minimum. 


EXTENSION  OF  SARRUS'S  METHOD.  485 

421.  To  give  a  more  comprehensive  view  of  the  method 
of  M.  Sarrus  in  the  treatment  of  double  integrals,  we  now  pro- 
ceed to  a  more  general  problem.  But  the  reader  who  desires 
may  omit  the  discussion  of  the  following  example. 

Problem  LXV. 

//   is    required    to    maximize    or    minimize    the    expression 

U:=  I      I     Vdy  dx,  where  V  is  any  function  of  x,  y,  z,  p,  q,  r,  s, 

and  t. 

It  is  evident  that,  supposing  the  limiting  values  oi  x  and  j 
to  be  also  variable,  we  shall  have 

dU=  r^  r^VDydx+T"  r^VDxdy 

+  Vr^r-^  Vs<^s  +  Vt^t]dydx  =  o,  (i) 

Now  all  the  terms  except  the  last  three  are  to  be  trans- 
formed and  arranged  as  in  equation  (15),  Art.  397,  so  that  we 
have  to  consider  these  three  terms  only. 

422.  By  equation  (3),  Art.  377,  we  have 

*yxo  tyyo  cix 


486  CALCULUS  OF    VARIATIONS. 

Moreover,  by  equation  (4),  Art.  378,  we  have 


+eAo     /      W'^^-^^^-^+eAo     /     y^^y^'^r)^^^^^'  (4) 

PXi    tyo  px^    lyo 

-i.   /     ^^yVrSqdx-J^^   I  y,{y,Vr),Szdx.  (5) 

Now  we  must  observe  that  every  y^  refers  to  the  contour 
only,  and  hence  it  varies  with  x,  but  is  independent  of  the 
general  values  of  jf.     Hence 

{Vry.y  =  Vry,.^  F,%        (F,j,X  =yxVrr  (6) 

Substituting  these  values  and  collecting  results,  we  may  write 

L    L     Vr^rdydx^J^^    l^     \Vry..+  2Vr%+Vr,{y.r\Szdx 

+X   L   Vr{y.rSgdx-l^   J^^   Vr'S.dy+l^  l^    V^Spdy 

-L   L    ^'ry.^'^L  Jy,    Vr'Szdydx.  (7) 

Again,  by  equation  (3),  Art.  377,  we  have 
rf^v/l?-dydx  = 

n^x     PVx  /«i     PVx  /»^i    fVx 

-L  X    V^'S'^dydx  +  l^  l^    VMdy-l^   l^    V.y.Sgdx.{Z) 


Kio) 


EXTENSION  OF  SARRUS' S  METHOD.  487 

By  equation  (5),  Art.  379,  we  have 

X'X"  ^-'^''y  '^^  -£:  ly^'^d''  ■'     (9) 

and  by  equation  (6),  Art.  380,  we  have 

/'X>-.f*=-/T'"..''*+/7."".'- 

-/'•r'-.f*=/"x>.."*-/7:''.*- 

Hence,  collecting  results,  we  have 

Lastly,  by  equation  (5),  Art.  379,  we  have 

-  Vt,-^-dydx^         /     Vt.Szdydx-         I    Vt^^zdx. 

Whence 


488  CALCULUS  OF  VARLATLONS. 

Adding  these  results,  and  also  the  second  member  of  equation 
(15),  Art.  397,  we  finally  obtain 


+  /     /    VDydx-\-        /     VDxdy    ' 

4-23.  Now  when  ^is  to  be  a  maximum  or  minimum,  we 
must,  as  before,  have  M  —  o  irrespectively  of  the  conditions 
which  are  to  hold  at  the'  limits.  This  equation,  which  must 
subsist  for  every  point  of  the  required  surface,  will  be  in  gen- 
eral of  the  fourth  order,  and  its  solution,  when  any  exists,  will 
not  contain  more  than  four  arbitrary  functions. 

Next,  if  in  the  terms  at  the  limits  we  regard  the  quantities 
Sz,  Sp^  dq,  Dy  and  Dx  as  independent  at  each  limit,  we  shall 
evidently  obtain  the  following  system  of  equations : 


(14) 


Vry„  +  Vr,{yif  +  (2  F/  -  V^)y^  +V,-V/-Vt,  =  o, 

Vr(.yxY-Vsy:r+Vt=0, 

Vp-V/-Vs,  =  o,        F,  =  o,  (15) 

F.-F-^.  =  o.  (16) 


V^o,        J^^   Vdy=o:  (17) 


EXTENSION  OF  SARRUS'S  METHOD.  489 

where  (14)  and  the  first  of  (17)  hold  along  the  upper  and  lower 
contour,  (15)  and  the  second  of  (17)  along-  the  left  and  right 
portions  when  they  exist,  while  (16)  holds  only  for  the  four 
corners,  or  the  junction  of  the  different  portions  of  the  con- 
tour, the  differentials  jFa:,  etc.,  at  these  points  being  taken  with 
reference  to  that  one  of  the  two  intersecting  portions  which 
we  may  happen  to  be  considering. 

But  under  the  present  supposition  the  total  number  of 
equations  at  the  limits  would  be  (16),  whereas  we  have  at  the 
most  not  more  than  four  arbitrary  functions  with  which  to 
satisfy  them  ;  so  that,  as  before,  we  must  impose  some  restric- 
tion upon  the  contour  which  will  reduce  the  number  of  these 
equations. 

424.  If  we  suppose  the  form  of  the  projected  contour  to 
be  fixed,  equations  (17)  will  disappear,  and  we  shall  have  but 
twelve  equations  at  the  limits ;  and  if,  in  addition,  we  suppose 
the  left  and  right  portions  to  be  wanting,  equations  (15)  and 
(16)  will  also  cease  to  exist,  and  we  shall  have  but  four  equa- 
tions at  the  limits.  In  this  case,  therefore,  in  which  \}i\Q  pro- 
jected contour  consists  merely  of  two  curves  which  meet,  we 
may  reasonably  suppose  that  a  complete  solution  might  be 
possible. 

We  can  render  equations  (14)  somewhat  more  symmetrical. 
For  differentiating  the  second,  regarding  7  as  a  function  of  x, 
as  indeed  it  is  along  the  projected  contour,  we  obtain 

+  {Vt-Vsy.+  V/  =  o.     (18) 

Now  from  this  equation  we  eliminate  f^x  by  the  first  of  equa- 
tions (14),  obtaining  an  equation  involving  jx  with  its  second 
and  third  powers.  Then  from  this  new  equation  eliminate 
successively  (jxT  and  (fa^Y  by  means  of  the  second  of  equations 
T4),  the  work  presenting  no  difficulty  whatever,  except  its 


490  CALCULUS  OF  VARLATLONS. 

length.      By  these  operations,  and  retaining  the  second  of 
equations  (14),  we  shall  have 


-^\VtVr,^VlV^-V,-Vr')^Vr{V,'^ZV,-2V^)\dy^o, 

Vrdf-  V,dydx-\-  Vtdx'^o. 


(19) 


4-25.  It  is  easy  to  show  that  conditions  (14),  or  rather  (19), 
must  hold  also  along  the  right  and  left  portions  of  the  con- 
tour when  they  exist.  For  since  along  these  portions  dx  =  o, 
the  second  of  equations  (19)  will  give  Fi-  ==  o ;  so  that  F^^  =  o, 
and  then  these  three  conditions  will  cause  the  first  of  equa- 
tions (19)  to  reduce  to 

Fp-F/-F,,=:o. 

For  the  four  corners  of  the  required  surface  we  merely 
join  to  equations  (14)  or  (19)  equation  (16). 

We  might  in  the  same  manner  as  before  discuss  the  case 
in  which  the  required  surface  is  to  be  limited  by  any  given 
surface  or  surfaces.  But  as  this  examination  would  not  prove 
useful,  because  of  the  scarcity  of  actual  problems,  and  as  it  is 
believed  that  the  reader  will  now  be  able  to  investigate  these 
cases  for  himself,  we  shall  proceed  no  further  in  the  discussion 
of  this  subject. 

426,  We  have  now  seen  that  the  method  of  M.  Sarrus 
enables  us  to  investigate  in  a  systematic  manner  the  condi- 
tions which  must,  under  any  supposition,  'hold  at  the  limits  in 
order  that  U  may  be  a  maximum  or  a  minimum ;  and  so  far 
as  this  method  itself  is  concerned,  it  should  be  regarded  as 
satisfactory  and  sufficient.  But  while  it  gives  the  conditions 
which  must  prevail  at  the  limits,  if  there  be  any  solution,  it 
still  remains  for  us  to  determine  whether  or  not  these  condi- 
tions can  be  fulfilled,  and  we  shall  find  at  this  point  that  the 


EXTENSION  OF  SARRUS'  METHOD.  .         49 1 

theory  is  much  less  satisfactory  than  in  the  case  of  simple  in- 
tegrals. For  supposing  V  to  contain  differential  coefficients 
of  z  to  the  order  n  inclusive,  we  know  that  the  equation  M^o 
will  be  in  general  a  partial  differential  equation  of  the  order 
2n.  Now  we  are  rarely  able  to  integrate  an  equation  of  this 
class,  and  are  not  certain  that  all  such  equations  admit  of  any 
solution  at  all  in  finite  terms ;  and  even  if  we  suppose  a  solu- 
tion to  exist,  we  cannot  tell  a  priori  how  many  arbitrary  func- 
tions it  must  involve,  all  that  we  know  being  that  the  number 
of  these  functions  will  not  exceed  that  which  marks  the  order 
of  the  differential  equation  in  question.  Moreover,  even  if  we 
knew  the  number  of  these  functions,  we  could  not  say  how 
many  conditions  ihey  might  be  made  to  satisfy,  since  we  would 
not  know  what  should  be  the  quantities  under  the  functional 
sign.  Also,  when  we  have  obtained  an  integral  of  one  of  these 
equations,  we  cannot  be  always  certain  that  the  solution  is  of 
the  most  general  possible  character. 

427.  From  what  has  been  said,  it  will  appear  that  we  can- 
not, as  in  the  case  of  simple  integrals,  assert  that  because  the 
equation  iI/=  o  is  of  the  order*  2??,  the  general  solution  can 
be  subjected  to  2n  conditions  at  the  limits  ;  although  the  ex- 
amination of  particular  cases,  as  well  as  the  analogy  of  simple 
integrals,  would  lead  us  to  infer  such  to  be  the  case.  If,  for 
example,  we  require  that  the  surface  given  by  the  equation 
M  —  o  shall  pass  through  2n  distinct  curves,  or  shall  have  its 
edges  upon  2n  surfaces,  we  do  not  know  that  these  conditions 
can  be  satisfied,  but  our  inference  that  the}^  can  is  supported 
by  the  following  additional  considerations. 

In  an  equation  of  the  form  M  =  o  we  can  assign  arbi- 
trarily the  values  of  z  corresponding  to  x  =  o  or  to  some 
function  of  x  and  y  equals  zero,  and  also  those  of  the  first 
2n  —  I  differential  coefficients  of  z  with  respect  to  either  vari- 
able, X  for  example.  Now  by  assigning  the  values  of  z  we 
compel  the  surface  to  pass  through  one  given  curve,  which 


492  CALCULUS  OF  VARIATIONS. 

would  be  all  that  we  could  do  in  the  case  of  a  partial  differ- 
ential equation  of  the  first  order.  When  the  equation  is  of 
the  second  order,  we  can,  as  before,  make  the  surface  first 
pass  through  some  curve,  and  then,  by  suitably  assigning  the 
value  of/,  can  fix  the  position  of  the  tangent  plane  along  this 
curv,e ;  that  is,  can  make  the  surface  pass  through  two  curves 
which  are  consecutive. 

In  like  manner,  when  the  equation  is  of  the  order  2n,  we 
can  effect  that  the  surface  shall  pass  through  2n  curves  which 
are  consecutive  one  to  another ;  and  since  this  can  be  done  so 
long  as.  the  curves  are  indefinitely  near  one  another,  we  may 
infer  that  it  would  also  be  possible  if  the  curves  were  sepa- 
rated by  finite  spaces,  although  we  must  be  careful  not  to 
speak  with  too  much  certainty  upon  this  point. 

The  last  two  articles  are  due  chiefly  to  Moigno  and  Lin- 
delof.     See  their  Calciil  des  Variations. 

428.  The  equation  M  =  o  will  not,  however,  always  rise 
to  the  order  2n.  If  Fbe  a  function  involving  x,  y,  z,p  and  q 
only,  thus  naturally  making  M  oi  the  second  order,  it  is  readily 
shown  that  M  will  not  rise  above  the  first  order  if  V  have  the 
form 

V  =  fix.y,  z)  +flx,y,  z)p  +flx,y,  z)q,  (i) 

and  in  this  case  only.  But  if  F  contain  x^y,  z,p,  q,  r,  s  and  /, 
giving  usually  M  oi  the  fourth. order,  it  can  be  shown  that  to 
prevent  M  from  rising  above  the  third  order,  it  is  necessary 
and  sufficient  that,  A,  B,  C,  D  and  E  being  severally  functions 
of  x^y,  z,p  and  ^,  F  shall  be  of  the  general  form 

V=.A(rt-s')^Br-\-2Cs-^Dt-^E.  (2) 

Moreover,  it  is  shown  that  in  both  these  cases  the  equation 
M  —o  cannot  in  reality  rise  above  the  order  2n  —  2.  See  the 
work  of  Prof.  Jellett,  page  249. 

It  will  be  remembered  that  the  corresponding  case   for 


EXTENSION  OF  JACOB  PS   THEOREM.  493 

simple  integrals  arises  from  the  fact  that  the  integral   /   ^Vdx 

is  capable  of  some  reduction  by  integration,  and  should  be 
reduced  before  applying  the  calculus  of  variations.  But  we 
cannot  extend  the  analogy.  For  in  the  present  case  no  such 
reduction  is,  in  general,  possible. 


Section  IV. 

EXTENSION  OF  JACOBPS  THEOREM  TO  THE  DISCRIMINATION 
OF  MAXIMA  AND  MINIMA   OF  DOUBIE  INTEGRALS. 

429,  We  will  now  present  a  mere  outline  of  the  method 
of  extending  the  theorem  of  Jacobi  to  double  integrals,  con-      / 
sidering  the  case  in  which  F  is  a  function  of  x,  y,  z,  p  and  q 
only,  and  supposing,  as  usual,  that  the  limiting  values  of  x,  y 
and  z  are  fixed. 

Now  since  the  terms  of  the  first  order  must  vanish,  if  U  is 
to  become  a  maximum  or  a  minimum,  we  shall  have 

6U=-  r^  r^ \  V,M  +2  V,p6z Sp  +  2  F^3 Sz dq 

+2F^,(^/d^+Fpp^/+  V^^6q''\dydx,{i) 
Now  we  can  change  the  form  of  (J C/" thus: 

6U  = 

\£'J^^\V,,Sz+V,,6p+V,^6q-{V,,Sz+V^p6p+V^,dgy 

-  ( Vzq  Sz  +  Fpg  Sp  +  F^g  Sq)^  ]  Sz  dy  dx.  (2) 

The   truth   of   (2)   can  easily  be   verified  by  integrating 
once  by  parts  each  of  the  quantities  within  the  accented  pa- 


494  CALCULUS   OF  VARIATIONS. 

rentheses,  the  first  set  with  respect  to  x  and  the  second  with 
respect  to  y,  remembering  that  the  limiting  values  of  z  are 
fixed.    Thus,  for  example, 

Proceeding  thus  with  each  term,  we  shall  obtain  the  same 
form  for  (5' C/ as  in  (i). 
Now  let 

Then  (2)  may  be  written 

-  (  Vp<i^P  +  Vq<i ^^).  \  ^^  dy  dx,       (3) 

430.  Thus  it  will  appear,  upon  comparing  equation  (3) 
with  equation  (7),  Art.  129,  that  SU  has  been  put  under  a 
form  which  \ye  may  call  Jacobi's  form  for  two  independent 
variables.  Moreover,  it  will  appear,  as  in  the  case  of  simple 
integrals,  that,  because  the  limiting  values  of  ;r,  y  and  z  are 
fixed,  we  must  have 

^U='-£'£'SMS.dydx;  (4) 

SO  that  we  have 

m=BS,  -  (Fpp<y/+  V^^Sq)'  -  {V^^Sp^  V„Sg),        (5) 
Now  let  u  be  such  a  quantity  as  will  satisfy  the  equation 
Bu  -  (Fpp«'  +  V^^u)'  -  {V„u'  +  V,^ul  =  o.  (6) 

Then  if  Sz  can  be  made  equal  to  u  or  ku  throughout  the  whole 
or  a  portion  of  the  double  integral,  k  being  an  infinitesimal 


EXTENSION  OF  JACOBPS   THEOREM.  495 

constant,  SU  \.o  the  second  order  can  be  made  to  vanish,  and 
we  would  infer,  as  before,  that  U  is  neither  a  maximum  nor  a 
minimum. 

Now  in  (3)  put  ut  for  S3.     Then  the  resulting  equation 
may  be  written 

w=  \^But-\Vpp{uty+Vp^{ut)X-\Vpq{uty-{-Vq^{tit)^\^  |  u.{^) 

But  because  (6)  is  true,  W  is  integrable.  For,  multiplying 
(6)  by  ut  and  subtracting  from  f'Fas  in  Art.  135,  we  have 

-  j^=:  u{  Vppiutyy-  ut\  Vppu'Y+^{  v^^(ut)x-  ut\  v^^ux 

-\-u\  Vpq{tay]-ut\  V^q?c'},+  u{  V^^{ui)^\-ut\  Fg,?/J,.  (9) 

Now  proceeding  with  each  pair  of  terms  precisely  as  in  Art. 
135,  it  is  evident  that  the  first  and  last  will  give  no  trouble, 
and  we  shall  also  find  that  no  difficulty  will  occur  in  the  sec- 
ond, but  tt'''  and  u^  will  merely  be  replaced  by  z//.  Proceed- 
ing then  as  indicated,  we  shall  ultimately  find 

w^  -  {{v,,f+v^,0ux-{{v^,^'  +  v,,ty\,.      (10) 

Substituting  this  value  in  (7),  and  integrating  by  parts,  one 
portion  with  respect  to  x,  and  the  other  with  respect  to  j/,  ob- 

servins:  that  /  or  —  must  vanish  at  the  limits,  we  shall  obtain 
u 

* 

^  J.CX'S^^^'"+^^i'*''''+  Vmt:\^'dydx.  (ii) 


49^  CALCULUS  OF  VARLATIONS. 

431.  Now  it  will  first  of  all  be  necessary  to  the  existence 
of  a  maximum  or  a  minimum  that  the  coefficient  of  u^  shall  be 
of  invariable  sign  throughout  the  field  of  the  double  integral. 

t' 
Putting  T  for  -,  this  coefficient  may  be  written 
^/ 

vJ.r^2Y-pT^'^'i\T:.  (12) 

But  to  secure  that  the  middle  factor  of  (12)  shall  be  incapable 
of  changing  its  sign  or  vanishing  for  any  real  value  of  T  posi- 
tive or  negative,  the  equation 

must  be  incapable  of  being  satisfied  by  any  but  two  imaginary 
values  of  T;  so  that  we  must  have 

( ^pp)        ^pp 

Therefore  it  is  necessary  to  the  existence  of  a  maximum  or 
a  minimum  that  Vpp  shall  be  of  invariable  sign  throughout  the 
portion  of  the  double  integral  which  we  are  considering,  and 
also  that  Vpp  Vqq  —  ( Vpgf  shall  be  always  positive,  although  it 
may  vanish  at  some  point.  By  reference  to  works  on  the  dif- 
ferential calculus  it  will  appear  that  these  conditions  are  anal- 
ogous to  those  which  must  hold  when  we  seek  by  the  ordinary 
method  to  maximize  or  minimize  a  function  of  two  variables 
which  are  independent. 

432,  But  it  is  evident  that  before  we  can  assert  in  any 
particular  case  that  we  have  a  maximum  or  a  minimum,  we 
must,  after  finding  the  two  above  conditions  to  be  satisfdctory, 
be  able  to  show  that  u  ox  ku  is  not  an  admissible  value  of  Sz 
throughout  any  finite  portion  of  the  integral,  and  also  that  no 


EXTENSION  OF  JACOB rs   THEOREM.  497 

element  of  <^^will  become  infinite.  To  ascertain  these  points 
we  must  be  able  to  determine  the  quantity  u  ;  and  here  the 
theory  practically  fails,  although  in  theory  u  may  be  deter- 
mined in  the  following  manner : 

Suppose  the  equation  M  =  o  were  completely  integrable, 
the  integrals  being  of  such  a  form  that  we  could  obtain  z  as 
a  function  of  x  and  y,  and  probably  two  arbitrary  functions  of 
X  and  y,  and  then  that  by  means  of  the  conditions  furnished 
by  the  fixed  contour  z  could  be  found  as  a  known  function  of 
x^y  and  two  constants,  say  z=f{x,y,  c^,  c^  =/.  If  now  we 
vary  c^  and  c^,  the  corresponding  values  of  Sz,  dp  and  Sq, 
although  not  necessarily  zero,  will  be  such  that  z-\-  6z,  p-{-dp 
and  q -\-  dg  will  still  satisfy  the  equation  M  =  o\  that  is,  SM 
will  be  zero.  Therefore,  because  (5)  is  true,  it  will  appear,  by 
precisely  the  same  reasoning  as  in  Art  132,  that 

The  preceding  discussion  is  all  that  we  have  space  to  pre- 
sent, nor  would  a  more  extended  treatment  prove  profitable. 
But  the  reader  who  may  wish  to  pursue  this  subject  further 
is  referred  to  an  article  by  A.  Clebsch  on  the  reduction  of  the 
second  variation  of  a  multiple  integral,  contained  in  the  fifty- 
sixth  volume  of  Crelle's  Mathematical  Journal  iox  1859. 


49^  CALCULUS  OF  VARIATIONS, 

Section  V. 

MAXIMA   AND  MINIMA    OF   TRIPLE  INTEGRALS. 

Problem   LXVI. 

433.  Let  u  be  the  density  at  any  point  of  a  body  whose  form, 
position  and  mass  are  known.      Then,  denoting  by  p,  q  and  r  the 

partial  differentials  — •,  — -  and  -— ,  it  is  required  to  determine 
•^  dx    dy  dz 

the  law  of  this  density,  so  as  to  minimize  the  expression 

z*^!   f*y\   n^x    , 

XXX    ^f^^f  +  't^-r'd.dydx.  (I) 

Since  the  mass  of  the  body  is  to  remain  constant,  we  must 

have 

rx^  pvi  nzx 

III     udzdydx  .  (2) 

Oxq    vfyo    t/Zo  -^  ^   J 

always  constant. 

Now  extending  the  method  of  double  integrals,  we  always 
suppose  that  when  u  is  known  as  a  function  of  x,  y  and  z,  (i) 
and  (2)  are  first  integrated  with  reference  to  z  only,  x  and  y 
being  regarded  as  constants ;  and  for  this  purpose  we  must  first 
substitute  in  (2)  the  value  of  ?^  as  a  function  of  x,  y  and  z,  and 
in  (i)  the  values  of/,  q  and  r  derived  from  this  function.  In 
other  words,  the  body  is  supposed  to  be  divided,  by  planes 
parallel  to  the  co-ordinate  planes,  into  prisms  whose  edges 
are  dz,  dy  and  dx ;  and  we  first  sum  up  these  prisms  along  any 
ordinate  z,  or  we,  at  any  rate,  obtain  the  portion  of  the  integral 
comprised  within  this  column. 

Thus,  considering  (2),  for  example,  we  would  have 

dy  dx  j   \idz  ^=^  /f{x,y,z)dydx,  (3) 


TRIPLE  INTEGRALS.  499 

which  will  give  us  the  solidity  of  any  right  parallelepipedon 
whose  edges  are  z^  —  z^^  dy  and  dx. 

Before  we  can  proceed  further  with  the  integration,  z^  and 
z^  must  be  determined  as  functions  of  x  and  y ;  that  is,  the 
body  must  be  limited  in  the  direction  of  the  ^'s  by  surfaces 
whose  equations  are  z  =  z^  and  z  =  z^,  z^  and  z^  being  known 
functions  of  x  and  y.  After  this  substitution,  (3)  may  be 
written 

dydx  I  ^udz  =  f'{x^  y)  dy  dx,  (4) 

We  next  integrate  with  reference  to  y  only ;  that  is,  we  sum 
up  all  the  parallelepipedons  corresponding  to  any  particular 
value  of  x^  and  thus  obtain 

which  will  give  us  the  solidity  of  a  section  of  the  thickness 
dx,  and  cut  from  the  body  by  two  planes  at  right  angles  to  x. 

Now  the  values  y^  and  y^  corresponding  to  any  particular 
value  of  X,  are  the  limits  of  this  section  in  the  direction  of  the 
ys,.  But  instead  of  supposing  any  section  to  terminate  in  a 
point,  we  shall  suppose  it  to  be  terminated  by  a  right  Hue  per- 
pendicular to  the  plane  of  xy,  because  this  supposition  is  more 
general,  the  former  being  at  once  deducible  from  it  by  merely 
reducing  these  terminal  lines  to  points. 

Hence,  under  the  most  general  supposition,  the  body  is  sup- 
posed to  be  limited  in  the  direction  of  the  j/'s  by  certain  cylin- 
drical walls  whose  equations  are  y  —  y^  and  j  =  y^ ;  and  there- 
fore before  integrating  again  we  must  determine  y^  and  y^  as 
functions  of  x,  and  substitute  this  value  in  (5).  Then  (5)  may 
be  written 

dx  j      j    \  dz  dy  =  F\x)  dx,  (6) 


500  CALCULUS  OF  VARIATIONS. 

We  now  integrate  with  respect  to  x ;  that  is,  sum  up  the  sec- 
tions just  mentioned.     This  will  give 

r^  r"  r\  dz  dy  dx  =^  r^F'dx = r'F"{x\     (7) 

Now  the  most  general  supposition  is  that  the  body  is  ter- 
minated in  the  direction  of  the  ^s  by  two  planes  perpendicu- 
lar to  X,  whose  equations  are  x  ^^^  x^  and  x  ^=^  x^.  For  other- 
wise it  can  only  terminate  in  an  edge  perpendicular  to  x,  or 
in  a  point,  both  of  which  cases  are  at  once  deducible  from  the 
first  supposition. 

4-34.  Thus  as  a  geometrical  conception  we  may  consider 
any  definite  triple  integral  as  extending  throughout  the  entire 
space  comprised  within  six  faces ;  the  first  two,  z  =^  z^  and 
z  =  z^,  which  we  shall  denote  by  Q  and  C^,  being  of  any  char- 
acter whatever,  either  face  being,  if  necessary,  made  up  of  sur- 
faces satisfying  different  equations ;  the  second  two  being  the 
cylindrical  walls  y  =  ^o  ^^d  y  =  j^,  which  we  shall  denote  by 
B^  and  B^,  either  face  being  at  liberty  to  become  merely  a  con- 
tinuous or  discontinuous  edge,  or  to  be  composed  of  different 
cylindrical  faces  whose  generatrices  are  parallel  to  the  axis 
of  z;  the  third  two,  which  we  shall  denote  by  A^  and  A^,  be- 
ing merely  the  planes  x  =  x^  and  x  —  x^,  where  either  plane 
may  reduce  to  a  point  or  to  any  right  line  perpendicular  to  x. 

435.  Now  suppose  that  throughout  the  solid  given  by  (2) 
we  make,  according  to  some  law,  an  infinitesimal  change  in 
the  density  u.  Then  we  shall  obtain  a  new  solid  which,  while 
not  differing  from  the  first  in  form,  will  differ  in  its  molecular 
condition,  and  may  be  called  the  derived  solid.  Moreover, 
to  obtain  the  difference  in  the  masses  of  these  solids,  we  have 
merely  to  sum  up  the  changes  which  take  place  in  each  ele- 
ment udz  dy  dx. 

But  in  varying  any  element,  it  is  most  natural  to  consider 


TRIPLE  INTEGRALS.  50I 

the  parallelepipedon  as  undergoing  no  change  whatever  in 
position  or  form,  but  in  density  only.  That  is,  we  may  regard 
X,  y  and  z,  and  consequently  dx,  dy  and  dz,  the  edges  of  the 
parallelepipedon,  as  incapable  of  variation.  Hence,  the  mass 
being  ^n^  we  would  have 


6m  =  III  ^^u'dzdydx, 

t/arc    t/t/o    e/zo 

Now,  to  generalize,  let 

f*^i     PVi     f*Zx 

U=         /      /     Vdzdydx, 

tl^O     t/i/o     *^Zo  -^ 


where  V  is  any  function  of  x,y,  z,  u,J),  q  and  r.  Then  it  will 
appear,  by  the  same  reasoning  as  before,  that  when  we  vary 
u,  p,  q  and  r,  the  limiting  values  of  x,  y  and  z — that  is,'  the 
limiting  faces — being  fixed,  the  corresponding  change  in  U 
will  be 

^U=S.l'  Sy!  S.y^-^^  +  V^^P^V-i^1^Vr6r\  dzdydx.    (8) 

Moreover,  it  is  evident  that  if  V  contain  differential  coefficients 
to  any  order,  the  same  method  must  be  pursued  in  obtaining 
6U. 

436.  Now  let  us  first  assume  u  to  undergo  no  variation  at 
the  limits ;  that  is,  along  the  six  faces.  Or,  to  fix  our  ideas,  sup- 
pose that  in  equation  (i)  or  (2)  the  density  were  required  to 
remain  fixed  throughout  all  the  limiting  faces,  or  the  surface 
of  the  body,  it  being  assigned  by  us  arbitrarily  for  each  limit- 
ing surface  at  the  outset.  Then  6u  will  vanish  at  the  limits, 
and  it  is  evident  that  by  integrating  by  parts  we  can  obtain, 
from  (8), 

^u=rrr  k-  -  ^  -  ^  -  ^  i  ^^^^^^y^^- 

t/o-o  t/yo  dz^     {  dx         dy         dz   ) 


r^\  nvx  pzi 
/      /      /     MSu  dzdydx. 

dXa     v'Va     fJZn  "^ 


S: 


(9) 


502  CALCULUS  OF    VARIATIONS. 

In  like  manner  we  may  treat  the  case  in  which   V  contains 
differential  coefficients  of  u  to  any  order. 

4-37,  Returning-  to  (8),  let  us  now  consider  how  to  trans- 
form the  variation  of  U  when  u  is  unrestricted  at  the  limits. 
By  equation  (7),  Art.  382,  we  have,  putting 


dt 
dx 


du  =  t,         dp  =  -J-,  Vp  =  u. 


nx\   pvi    PZi 
X  X  X    V^Spdzdydx  = 

Jxq    tlyo    tJzo       dx  '^0    ^^2/0    t/2;o        ^ 

-  rr  rvZ-f^-dzdx-rrr'v/-^  sudydx.  (lo 

t/a^o     /2/0    t/^^o        ^  dx  *^^o    *^^o     '^0  dx 

Again,  writing  F^  =z  u,  Sq  =  — ,  we  have,  by  (8),  Art.  383, 

r  r  rv,sqdzdydx=-rrr'^-^sudzdydx 

-^  r  /"'  pV,6udzdx-rr  rv/Uudydx.{.i) 

^  uxo    ivq  Jzo       3  Jxo  *Jyo    Iz^      ^  dy 

Lastly,  by  equation  (9),  Art.  384,  or  by  integrating  directly 
by  parts,  we  have 

'      /      /     Vrdrdzdydx  = 

Xq      t/T/o      *^Z0 

-  r  r  n^j^sudzdydx+rr r^sudydx.  (12) 


TRIPLE   INTEGRALS.  503 

SuDStituting  these  values  in  (8),  and  arranging,  we  have 

+rrr{^'-f-f-^^i-^.*-  (.3. 

In  this  form  of  c^^^we  observe  that  there  are  four  terms,  the 
first  relating  to  the  faces  C^  and  C^  only ;  the  second  to  the 
cylinders  B^  and  B^  only;  the  third  to  the  planes  A^  and  A^ 
only ;  while  the  fourth  extends  throughout  the  entire  inte- 
gral U. 

438.  Now  when  U  is  to  be  a  maximum  or  a  minimum, 
SUto  the  first  order  must  vanish  ;  and  it  is  evident  that  whether 
the  terms  at  the  limits  exist  or  not,  we  shall  obtain  the  equa- 
tion 

an  equation  which  holds  throughout  the  integral.  The  inte- 
gral of  this  equation  will  involve  certain  arbitrary  functions ; 
but  if  //  be  invariable  at  the  limits,  the  calculus  of  variations 
affords  us  no  further  equations  of  condition,  and  these  func- 
tions must  be  determined  by  the  values  which  we  require  11 
to  maintain  upon  the  various  faces  which  limit  the  integral. 

But  when  u  is  unrestricted  at  the  limits,  we  must  also  have 
L  —  o,  L  denoting  the  terms  at  the  limits  ;  and  since  6u  (for 
example,  the  variation  of  the  density,  if,  as  in  (i)  and  (2),  u  be 
the  density)  is  entirely  in  our  power  for  each  point  of  the 


504  CALCULUS  OF    VARIATLONS. 

various  faces,  these  faces  themselves  undergoing  no  change  in 
form,  it  is  easy  enough  to  see  that  we  must  have  the  equations 


^-^^^-^'1-°' 


dy 
dx 


V,-  F^^  =  o,    Fp 


(15) 


the  first  equation  holding  for  the  faces  Q  and  C^,  the  differen- 

djs  dz 

tials  -r-  and  --  being  relative  to  these  faces  only;  the  second 
dx  dy 

dv 
holding  for  the  walls  B^  and  B^,  the  differential  -j-  being  rela- 

dX 

tives  to  these  walls  only ;  and  the  third  holding  only  through- 
out the  planes  A^  and  A^, 

439,  We  will  now  show  that,  as  in  the  case  of  double  in- 
tegrals, these  conditions  are  in  reality  identical. 

Let  a^  b  and  c  denote  the  angles  made  with  the  co-ordinate 
axes  by  any  normal  to  C^  or  C^.  Then  the  first  of  equations 
(15)  will  give  the  conditions 

,  Vp  cos  a-\-  Vq  cos  b-\-  Vr  cos  c  =  0.  (16) 

Now  for  the  cylindrical  faces  B,  and  B,  we  have  cos  c  =  o, 
and  the  second  of  equations  (15)  would  therefore  give  the 
condition 

Vp  cos  a-\-  Vq  cos  b  =  0,  (17) 

which  would  follow  at  once  from  (16)  when  cos  <:,  vanishes. 
Lastly,  for  the  planes  A^  and  A^,  we  have  cos  r  =  o,  cos  b  —  o 
and  cos  a  —  i-,  so  that  equation  (16)  would  give  at  once 
Vp  =  Q,  the  equation  required.  It  appears,  therefore,  that  the 
first  of  equations  (15),  or  rather  that  equation  (16),  holds  for 
all  the  faces. 


TRIPLE  INTEGRALS.  505 

440.  Now  it  is  clear  that  the  problem"  proposed  at  the 
beginning  of  this  section  is  one  of  relative  "minima,  since  the 
variations  of  u,  -p,  q  and  r  are  to  be  so  restricted  as  to  permit 
a  certain  mass  to  remain  constant.  But  it  will  readily  appear 
that  by  selecting  portions  of  the  solid,  just  as  we  did  of  the 
curve  for  single  integrals  and  of  the  surface  for  double  in- 
tegrals, we  can  extend  the  method  of  Euler,  by  the  reasoning 
of  Bertrand,  to  triple  integrals  also ;  and  we  shall  therefore 
assume  this  fact  without  further  discussion. 

Now  assuming,  for  convenience, as  the  constant  multi- 

plier,  we  must,  as  we  see  from  (i)  and  (2),  minimize  the  ex- 
pression 

P^i     /*2/i     /*2i 

=  Vdzdydx.  (i8) 

Here 

I 


v^  = 


so  that  the  equation  M  —o  gives 

I    I    ^  P  

g       dx   Vi-^-p'+q'-^r'   '  dy    Vi -\- p"  +  q' +  r" 

■      ^  ^  =0.  (20) 


y,: 

~    4/1 

+/+f 

+  .'' 

Vr-- 

r 

V'l 

+/+?' 

+  r^ 

fa 
■A- 

d 

g 

\{i9) 


dz  Vi+/+/+^' 


According  to  Moigno,  one  solution  of  this  differential  equa-" 
tion  is 

{x  -  Kf + (J/  -  /r + (^  -/r+  (« -  ky  -  9^s     (21) 

//,  /,  j  and  k  being  constants. 


506  CALCULUS  OF  VARLATIONS. 

This  is  an  equation  which  is  analogous  to  that  of  the  circle 
and  sphere.  If  we  suppose  u  fixed  at  the  limits,  we  must  as- 
sign it  so  as  not  to  conflict  with  this  equation.  These  condi- 
tions could  evidently  all  be  fulfilled  if  the  body  were  a  sphere 
the  density  of  whose  surface  were  to  remain  uniform  through- 
out. 

When  ti  is  not  restricted  at  the  surface  of  the  body,  equa- 
tion (i6)  will  give  at  once  the  condition 

/  cos  a-\-  q  cos  b-{-r  cos  c-=o,  (22) 

which,  as  we  have  seen,  must  hold  for  all  the  limiting  faces. 

441.  Let  us  now  suppose  that  while  the  mass  of  the  body" 
is  to  remain  constant,  its  form  is  not  fixed,  its  density  at 
the  surface,  however,  being  required  always  to  satisfy  the 
equation 

f{x,y,z,u)=f-=o,  (23) 

/  being  any  function  whatever. 

This  case  corresponds,  for  the  triple  integral,  to  that  of  a 
double  mtegral  in  which  the  required  surface  is  to  have  its 
contour  always  upon  one  or  more  given  surfaces.  For  here 
the  faces  C^,  etc.,  take  the  place  of  the  contour,  u  is  the  de- 
pendent variable  instead  of  z,  and  the  hmiting  function  or 
functions  /  =  o  which  71  is  to  satisfy  upon  the  faces,  take  the 
place  of  the  equation  or  equations  of  the  surface  or  surfaces 
upon  which  the  contour  must  rest. 

But  to  enable  us  to  discuss  this  case,  we  must  first  con- 
sider how  to  find  the  variation  of  U  when  the  limiting  values 
of  X,  y  and  z,  as  well  as  those  of  u,  are  variable,  where 


UXq      t/Va      tJZn 


V  being  any  function  of  x,  y,  z,  u,  p,  q  and  r.     We  can  evi- 
dently pass  from  the  primitive  to  any  derived  solid  by  first 


TRIPLE  INTEGRALS.  50/ 

varying  the  limiting  faces,  supposing  u  to  remain  unvaried, 
and  then  varying  u  throughout  the  new  sohd.  In  var3dng 
the  faces,  suppose  first  that  x  and  y  remain  constant,  and  that 
we  change  any  two  ordinates  z^  and  z^  of  the  faces  C^  and  C^ 
into  ^0  +  ^-^0  ^nd  z^  -\-  Dz^.  Then,  by  precisely  the  same  rea- 
soning as  if  z  were  the  only  independent  variable,  as  indeed  it 
is  so  long  as  we  are  passing  from  z^  to  z^  only,  we  shall  have, 

to  the  first  order,  as  the  change  in  /   '  Vdz,  Vfiz^  —  V^Dz^ ; 

and  extending  this  method  to  every  value  z^  and  z^,  we  shall 
have,  as  the  part  of  ^^  resulting  from  the  variation  of  the 
faces  C^  and  C^, 

We  may  next  vary  the  walls  B^  and  B^,  supposing  x,  z 
and  u  to  remain  unvaried.  But  in  varying  these  faces  we 
must  remember  that  they  are  always  to  remain  cyhndrical, 
every  generatrix  being  parallel  to  the  axis  of  z.  Now,  as 
before,  when  we  change  y^  into  y^  +  Dy^,  and  y^  into  y^  -[-  Dy^, 
X  and  z  remaining  unvaried,  y  is  the  only  independent  vari- 

able,  and  the  corresponding  change  in  /      Vdy  will,  to  the  first 

order,  be  V^Dy^  —  V^Dy^ ;  and  this,  being  taken  throughout  any 

/     VDydz,  where  Dy  is  a  constant 

throughout  the  generatrix  in  question,  but  independent  for 

each.    Now  since  we  can  only  vary  B^  and  B^  by  varymg  each 

generatrix  m  the  manner  described,  we  must  take  the  sum  of 

these  variations,  the  integration  being  with  respect  to  x\  and 

therefore  the  change  resulting  to  U  from  varying  B^  and  B^ 

will  be 

f*^i  ivi  r^\ 
X  /.  X   VDydzdx.  (25) 

We  next  vary  the  planes  A^  and  A^,  supposing  j/,  z  and  ti  to 
remain  unvaried,  and  keepmg  A^  and  A^  always  planes,  per- 


508  CALCULUS  OF    VARIATIONS. 

pendicular  to  the  axis  of  x.  Then  the  change  of  .r„  and  x^ 
into x^  +  Dx^  and  x^-\-Dx^  will  give  V^Dx^  —  V^Dx^.  Hence, 
extending  this  method  to  every  point  of  the  planes  A^  and^„ 
we  shall  obtain 


^'VDxdzdy,  (26) 


where  Dx^  and  Dx^  are  constants,  but  independent. 

Besides  those  which  we  have  evidently  omitted,  various 
other  terms  of  the  second  and  higher  orders  would  arise  at 
the  intersection  of  the  faces,  which  we  do  not  propose  to  con- 
sider. 

44-2.  If  now,  in  the  second  place,  having  varied  the  faces, 
we  vary  u  throughout  the  new  limits,  these  limiting  faces 
themselves  remaining  fixed,  it  is  evident  that  the  result  can- 
not differ  by  any  term  of  the  first  order  from  the  value  oi  SU 
before  the  limits  were  varied,  the  difference  consisting  only 
of  the  variations  of  (24),  (25)  and  (26),  themselves  quantities  of 
the  first  order  only.  Hence  this  part  of  SU  is  to  be  found 
and  transformed  as  already  explained.  Therefore,  finally,  add- 
ing (13),  (24),  (25)  and  (26),  we  shall  have 

Jx^    Jyo    Iz^      \      "^         P^^  ^  dy  S  "^ 

+  /     /     /     VDydzdx-\-        /      /     VDxdzdy 

+/VT"  I  ^»-  ^  -^  -^  I  ^ud.dyd.  =  0.  (27) 
Jx,   Jy,  ^zo     i     ""       ^j^         dy         dz    S  -^ 


TRIPLE  INTEGRALS.  509 

Moreover,  it  is  evident  that  if  V  contain  differential  coeffi- 
cients of  u  to  any  order,  we  shall  still  obtain  the  most  general 
value  of  dUhy  merely  adding  (24),  (25)  and  (26)  to  the  result 
obtained  by  varying  U^  supposing  the  limiting  values  of  x^  y 
and  z  to  be  fixed. 

44-3.  Now  if  U  is  to  become  a  maximum  or  a  minimum, 
we  must,  denoting  the  terms  at  the  limits  by  Z,  equate  L  and 
M  severally  to  zero,  where  M  is  the  same  as  before,  but  L  is 
not.  Then,  if  we  could  suppose  the  quantities  Dz,  Dy  and  Dx 
to  be  entirely  independent  of  Su^  w^e  shall  first  of  all  obtain 
from  L  —  Q  equations  (15),  besides  which,  if  Dz,  Dy  and  Dx 
be  independent,  we  must  equate  severally  to  zero  (24),  (25) 
and  (26).  But  since  Dz^  and  Dz^  have  all  the  independence  of 
variations,  (24)  will  give  F  =  o,  which,  together  with  the  first 
of  equations  (15),  must  hold  throughout  the  curved  surfaces 
C^  and  C^. 

Now  in  (25)  Dy  is  an  independent  constant  along  each  sev- 
eral generatrix,  so  that  each  element  of  the  integral  must  van- 
ish, and  we  must  have  /  VDydz  =  o,  which  holds  along  any 
single  generatrix  only ;  and  Dy  being  constant  along  this 
generatrix,  we  have  /     Vdz  =  o,  which  is  all  the  reduction 

we  can  effect,  and  must  hold  along  each  generatrix  of  the 
faces  B,  and  B,y  while  the  second  of  equations  (15)  holds  for 
each  point  of  these  faces. 

Lastly,  in  (26)  Dx^  and  Dx^  are  two  independent  constants, 

and  we  have,  therefore,  /      /     Vdzdy  =  o,  which  is  the  final 

equation,  and  holds  throughout  the  entire  planes  A^  and  A, 
only,  while  the  third  of  equations  (15)  must  hold  for  each  sep- 
arate point  of  these  faces. 

444.  Now  although  we  do  not  know,  a  priori,  just  how 
many  conditions  at  the  limits  the  solution  of  the  equation 


5IO  CALCULUS   OF  VARIATIONS. 

M=  o  can  be  made  to  satisfy,  the  number  will  probably  not 
exceed  two,  so  that  it  is  evident  that  the  quantities  du,  D2,  Dy 
and  Dx  cannot  be  regarded  as  entirely  independent,  since  this 
supposition  would,  as  we  have  seen,  give  us  twelve  equations 
at  the  limits,  two  relative  to  each  of  the  six  faces.  Hence 
we  must  impose  some  restriction  at  the  limits  of  the  kind  men- 
tioned in  the  beginning  of  Art.  441,  which  we  now  proceed  to 
consider,  but  at  first  in  a  general  manner,  and  not  relative  to 
any  particular  problem. 

445.  Suppose,  first,  that  upon  the  face  C^,  u  is  required  to 
equal  some  function  f  oi  x,  y  and  z,  this  face  itself  being  sub- 
ject to  variation  of  form ;  and  let  P,  Q  and  R  be  the  partial 
differential  coefficients  of  /  with  respect  to  x,  y  and  z.  Then, 
because,  when  we  pass  along  any  ordinate  z,  x  and  y  remain 
constant,  /  becomes  in  reality  a  function  of  z  and  constants 
only,  and  might,  therefore,  be  made  the  ordinate  of  a  plane 
curve,  z  being  the  abscissa.  Hence,  by  the  same  reasoning 
as  hitherto,  we  must  have 

l'"du=j'\R-r)Dz,  (28) 

and  a  similar  equation  for  the  lower  limit  of  z.  In  like  man- 
ner., if  we  suppose  that  u  must  equal /upon  each  of  the  other 
faces,  it  being  immaterial  whether  or  not  /  be  the  same  func- 
tion for  all  the  faces,  or  even  throughout  the  same  face,  we 
shall,  by  extending  to*;ir  and  y  the  reasoning  just  employed  for 
z^  obtain  for  B^  and  A^  respectively, 

5u=l    {Q-q)Dy,         /    6u=/     {P^p)Dx,      (29) 

with  similar  equations  for  the  lower  limits. 

Now  taking  the  value  of  L  from  {2^),  putting  together  the 
terms  affected  by  hke  signs  of  substitution,  and  then  elimi- 
nating du  from  each  by  equations  (28)  and  (29),  we  shall  have 


TRIPLE  INTEGRALS.  51I 

+/rxxs  ^+  ^.(^-/)s^--'-'^/ = o.       (30) 

446.  Now,  because  the  quantities  Dz^,  Dz^,  Dy^,  Dy^,  Dx^ 
and  Dx^  are  all  independent,  we  must,  in  the  first  place,  equate 
severally  to  zero  each  of  the  three,  or  rather  six  terms  in  L. 
Hence,  as  before,  the  first  will  evidently  give 

V^[Vr-v/^^-v/^(^R-r)^o,  (31) 

an  equation  which  relates  to  either  of  the  faces  C^  or  C^  only. 
Next,  by  the  same  reasoning  as  before,  the  second  term 
can  only  be  made  to  give 

X"  \V\[V,-V,  g)(G  -q)\dz^O,  (32) 

an  equation  which  holds  for  any  one  generatrix  only  of  the 
faces  B^  or  B^,  the  integral  being  required  to  be  taken  through- 
out that  entire  generatrix. 

Lastly,  the  third  term  will  give 

£lyV^V,(P-p)\dzdy  =  o,  (33) 

an  equation  which  holds  for  the  planes  A,  or  A,  only,  the  in- 
tegration being  required  to  extend  throughout  the  entire  sur- 
face of  either  plane. 

But  since  u  must  always  equal  /  upon  the  face  C^  or  C^,  if 


512  CALCULUS  OF  VARIATIONS. 

we  pass  from  one  point  to  another  upon  either  of  these  faces, 
the  change  which  u^  the  density,  for  example,  will  undergo, 
will  equal  the  corresponding  change  in  /,  however  these 
points  may  be  situated.  Let  us,  then,  assume  these  points  to 
be  indefinitely  near  each  other,  and  both  to  be  first  in  a  sec- 
tion cut  by  a  plane  parallel  to  that  of  xz,  and  then  in  one  cut 
by  a  plane  parallel  to  that  oi  yz.  These  suppositions  will  give 
respectively 

pdx -\-rdz  =:  Pdx -\- Rdz,         qdy -\- rdz  =  Qdy -\- Rdz, 

Whence 

^  -  _  ^-^  d^^_  Q-q 

dx~       R-r         dy  R-r  ^^"^^ 

By  similar  reasoning,  since  u  always  equals  /  upon  either 
cylindrical  wall  B^  or  B^,  if  we  pass  from  one  of  two  consecu- 
tive points  to  the  other  along  the  intersection  of  this  wall 
with  the  plane  of  xy,  we  must  have 

pdx  +  qdy  =  Pdx+Qdy,         g=-g^;  (35) 

which,  being  true  along  the  plane  of  xy,  is  of  course  true  for 
the  entire  face  B^  or  B^. 

Now  substituting  these  values  in  (31)  and  (32),  and  repro- 
ducing (33),  we  have 

F+  Fp(P  -  /)  +  V,{Q  -g)^Vr{R-r)  =  o, 

X''\V^V^{P-p)-^V^{Q-g)\d,  =  o,  y    (3g) 

4-4-7.  Such,  then,  are  the  equations  which  must  hold  for 
the  various  limiting  faces ;  and  the  reader  can  easily  apply 


TRIPLE   INTEGRALS.  513 

them  to  the  particular  problem  with  which  we  started, 
although  no  results  of  importance  present  themselves.  In- 
deed, we  assumed  this  problem  merely  because  it  better  fixes 
our  ideas  to  think  at  first  of  u,  the  dependent  variable,  as  some- 
thing physical  or  geometrical,  like  density,  than  as  some  func- 
tion merely  of  x,  y  and  z^  although  the  latter  view  will,  in 
general,  be  necessary. 

It  would  appear  that,  without  reducing  the  number  of  the 
limiting  faces,  we  shall  still  have  in  general  too  many  equa- 
tions at  the  limits,  although  our  imperfect  knowledge  as  to 
what  should  be  the  form  of  the  most  general  possible  solution 
of  the  equation  M=o  will  prevent  us  from  determining  how 
far  these  conditions  might  be  fulfilled. 

Although  the  converse  need  not  be  true,  the  second  and 
third  of  equations  (36)  would  be  satisfied  should  the  first  hold 
throughout  all  the  limiting  faces.  For  since  ?/=y  along  any 
particular  generatrix  of  the  ^'s,  R  and  r  must  be  always  equal 
along  that  generatrix ;  which  would  satisfy  the  second  equa- 
tion by  giving 

V+V^{P-p)  +  V,{Q-q)  =  o. 

In  like  manner,  because  u—f  throughout  the  planes  A^  and 
A^,  we  see,  by  first  passing  along  any  line  perpendicular  to 
the  plane  of  xy,  and  then  along  any  line  perpendicular  to 
that  of  xz,  that  throughout  the  A'%,  R  —  r,  Q  =  q,  and  the 
first  equation  satisfies  the  third  by  giving 

V+V^{P-p)  =  o, 

448.  When  the  limiting  values  of  x,  y,  z  and  u  are  all 
fixed,  the  terms  of  the  second  order  can  be  sometimes  ex- 
amined. Thus  in  the  particular  problem  with  which  we 
opened  this  section,  Vuu,  Vup,  Vuq  and  Vur  all  reduce  to  zero, 


SH  CALCULUS  OF  VARIATLONS. 

the  ten  terms  of  the  second  order  will  reduce  to  six,  and  we 
shall  easily  find  that  (J^may  be  exhibited  thus: 

+  {q^p-p^qf  +  {rdp  -pdrf  +  {rdq  -  qSrY]dzdydx, 

which  is  evidently  positive,  thus  giving  us  a  minimum. 

The  foregoing  discussion  will  render  the  reader  sufficiently 
familiar  with  the  treatment  of  triple  integrals,  while  a  discus- 
sion of  those  of  a  higher  order  would  be  of  no  use,  except  as 
a  matter  of  curiosity,  and  would  be  beyond  the  design  and 
scope  of  this  work. 


Section  VI. 

ANOTHER    VIEW  OF    VARIATIONS. 

449.  If,  in  the  preceding  discussion,  we  had  for  double 
integrals  ascribed  variations  to  x  and  y,  and  for  triple  to  x,  y 
and  ^,  we  could,  as  in  the  case  for  single  integrals,  have 
obtained  the  same  formulae  as  by  the  method  which  we  have 
adopted.  Or  we  might  even,  as  Prof.  Jellett  does,  in  the  case 
of  double  integrals,  assume  the  required  surface  as  the  inde- 
pendent variable,  considering  x,  y  and  z  as  functions  of  the 
surface. 

But  there  is  besides  these  another  more  analytical  view  of 
variations,  applicable  to  integrals  of  any  order,  which  pre- 
sented itself  to  Euler  and  Lagrange,  has  been  followed  essen- 
tially not  only  by  Strauch,  but  by  Sarrus,  and  subsequently 
by  Moigno  and  Lindelof,  as  will  appear  from  their  Calcul  des 
Variations^  Legon  III. 

450.  To  begin,  then,  with  the  simplest  case,  suppose  a 
plane  curve  whose  equation  is  r  =  f(x)  =:/,  and  change^  into 


ME  THOD  OF  PARA  ME  TERS.  5  I  5 

y  -\-  Sy  or  Y,x  remaining  unvaried.  Then  we  may  regard  Y 
as  a  function  F  oi  x  and  /,  where  /  is  a  new  quantity  entirely 
independent  of  x,  and  constant,  and  may  enter  F  in  any  man- 
ner we  please,  provided  only  that  the  form  of  F  shall  be  such 
as  to  cause  it  to  reduce  to /when  t  is  made  zero.  Then  it  is 
evident  that  if  we  regard  f  as  containing  t,  we  must  regard 
it  as  a  function  of  x  and  o.  Then,  since  x  does  not  vary  for 
any  change  in  /,  we  may,  by  Maclaurin's  theorem,  develop  Y 
in  ascending  powers  of  t,  obtaining 

''=/+'[f]+i[?]^-.     <■) 

where  brackets  denote  that  t  is  made  zero  after  the  differen- 
tial coefficients  of  F  with  respect  to  t  have  been  found.  If 
we  suppose  t  to  be  made  infinitesimal,  we  may  neglect  powers 
of  /  of  an  order  higher  than  the  first,  and  write 


^-'""-\S. 


t,  (2) 


in  which,  because  /  may  enter  F  in  any  manner  we  please 

which  will  cause  F  to  reduce  to  /  when  /  is  zero,     —      is 

entirely  in  our  power,  and  may  be  made  to  become  any  func- 
tion we  please.     Therefore,  replacing  F  by  y,  we  have 

where  dy  is  as  unrestricted  as  formerly. 

In  like  manner,  since,  when  jj/  =  /,  y  or  -^  z=f\x)  =/',  and 

when  y  is  supposed  to  be  F{x,  /),  y'  becomes  F\x,  t),  we  shall 
find 

V=[f]<.  (4) 


5l6  CALCULUS  OF   VARIATIONS. 

But  t  being  independent  of  x,  if  we  differentiate  (3)  with  re- 
spect to  X,  we  have 

dx  dxLdtJ  Ldx  dt  J         LdtJ 

Hence,  as  before, 

'^=?'  <« 

and  similarly  we  find  6/',  Sy'",  etc.,  to  be  as  usual. 

451.  Next  let  Fbe  any  function  of  x,y,y\y\  etc.  Then 
when  J/,  y ,  y\  etc.,  which  are  all  functions  of  x,  are  supposed 
to  become  such  arbitrary  functions  of  x  and  /  as  will  reduce 
them  to  their  original  values  when  t  is  made  zero,  V  must 
also  become  some  function  of  x  and  /,  and  we  have  at  once, 
as  in  the  case  of  y  and  y , 

-=K-]'=lf[|]+fB']+-.l' 


which,  to  the  first  order,  is  the  usual  form. 

Now  let  V  besides  y  contain  another  dependent  variable  z 
with  its  differential  coefficients  2',  z'\  etc.,  with  respect  to  x. 
Then  if  y  and  z  are  independent,  t  may  enter  either  in  any 
arbitrary  manner  which  will  permit  it,  and  also  its  differential 
coefficients,  to  remain  unchanged  when  t  is  zero.  But  if  y  and 
z  are  always  to  be  connected  by  an  equation,  differential  or 
other — that  is,  if  Sy  and  dz  are  to  be  related — t  may  enter  one 
of  these  quantities  in  any  arbitrary  manner,  but  must  enter 
the  other  in  such  a  way  as  to  cause  y  and  z  to  be  related  in 
the  required  manner.     In  either  case,  since  y  and  z  are  func- 


/ 


METHOD   OF  PARAMETERS.  51/ 

tions  of  X,  when  they  become  functions  of  x  and  /,  F  becomes 
also  a  function  of  x  and  / ;  and  proceeding  as  before,  we  have 

-=[f]-f[i]'+i;[f]'+- 


dV  r.     ,   dV  o.  ,   ,      .       ,   dV  ^     .   dV  j.  .  ,     ^         ,  . 

=w^^+w  ^^     ^^    '^^  '^+^'^-'  ^^^ 

which  is  the  same  form  as  before,  and  in  it,  as  we  have 
already  seen,  the  variations  may  be  independent  or  may  be 
made  to  fulfil  any  conditions  which  we  may  impose. 

In  like  manner,  if  Fbe  a  function  of  any  number  of  inde- 
pendent variables  x,  y,  z,  etc.,  and  a  dependent  variable  u  to- 
gether with  its  differential  coefficients  with  respect  to  x,  y,  z, 
etc.,  we  shall,  by  supposing  u  to  become  a -function  of  x,  y,  z, 
etc.,  and  /,  instead  of  x,  y,  Zy  etc.,  only, — t  being  entirely  inde- 
pendent of  these  variables, — render  Fa  function  of  x,y,z,  etc., 
and  /,  and  obtain 


where  the  second  member  will  always  be  the  same  as  we  would, 
by  the  ordinary  method,  obtain  as  the  variation  of  Fwhen  x, 
y  and  z  do  not  vary. 

4-52.  Now  suppose  we  have  the  equation  U  =J^    Vdx,  V 

being  any  function  of  x,  y,  y' ,  y",  etc.  Then  U  will  be  some 
function  of  x^  and  x^ ;  and  when  y  becomes  such  a  function  of 
X  and  /  as  will  reduce  Y  to  y  when  t  is  zero,  U  will  also  be- 
come such  a  function  of  x^,  x^  and  t  as  will  reduce  U-\-  SU  to 


5lS  CALCULUS   OF  VARIATIONS. 

U  by  making  t  zero.     Therefore,  proceeding  as  formerly,  we 
must,  to  the  first  order,  have 

where  — —  denotes  the  differentiation  of  U  with  respect  to  /, 

and  to  everything  which  in  any  way  depends  upon  /,  and 
nothing  more. 

Now,  in  the  most  general  case,  we  cannot  regard  the  limit- 
ing values  of  x  as  fixed,  but  we  may  suppose  these  also  to 
become  functions  of  t,  together  with  some  constants  indepen- 
dent of  /,  and  this  supposition  will  give  us  all  needed  general- 
ity.    In  (9)  the  increments  ^x^  and  Sx^  or  Dx^  and  Dx^  become 

/'■[§]' --m]'- 

Now  by  equation  (i),  Art.  375,  (9)  gives 

=^  r'svdx-^r^vDx, 

where  c^Fhas  the  form  given  in  (6),  and  S/^  dy\  etc.,  are,  as 
appears  from  (5),  capable  of  the  same  transformation  by  inte- 
gration as  usual. 

Moreover,  if  V  contain  the  dependent  variable  2  also,  and 
its  differential  coefficients,  the  last  equation  will  still  hold, 
only  SVwiW  take  the  form  given  in  (7),  and  may  be  trans- 
formed as  in  the  ordinary  case  of  two  dependent  variables ; 
so  that  for  single  integrals  we  always  obtain  at  once  the  same 


METHOD   OF  PARAMETERS.  519 

fundamental  and  the  same  limiting  equations  as  by  the  ordi- 
nary method. 

453.  The  reader  will  probably  now  be  ready  to  admit 
that  equation  (9)  must  hold  also  for  any  definite  multiple 
integral  whose  limits  are  fixed  or  variable. 

Now  in  equation  (i),  Art.  375,  change  u  into  /   \dy.   Then 

we  have,  by  the  aid  of  the  same  equation, 

—  /      /     udydx=. 


«/a:o    «/?/o      dt  ^^0     ho        dt  '^0    e/t/o         dt 

and  here  changing  u  into  /    u  dz,  we  obtain 
--/      /      /     udzdydx=. 
I      /      /      -—  dz  dy  dx-\-  /     /    u-—dy  dx 

^  Jxo    lyo   tJzo        df  '   Aco    t/2/0    «/2o        dt 

Hence  it  is  easy  to  see  that  we  would  have  respectively  for 
u—  I      I    '  Vdy  dx        and     u=  /      /    '  Vdz  dy  dx 

OXq    t/2/0  0x0    tJyo    t/2o 

the  two  equations 


520  CALCULUS  OF  VARIATLONS. 

6U=  r^  r^  r^ dV dz  dy  dx -\-  r^  r^  T'VDzdydx 

which  are  the  forms  previously  obtained  for  variable  limits, 
and  —  /ordFis  just  what  it  would  be  by  the  ordinar}^  meth- 
od, and  is  transformable  in  the  same  manner ;  so  that  here  also 
we  shall  obtain  at  once  the  same  general  and  the  same  limit- 
ing equations  by  either  method. 

4-54.  Before  proceeding  we  shall  require  some  additional 
formulae  in  the  calculus  of  substitution. 

In  (lo)  change  u  into  /  V.     Then  by  equation  (2),  Art.  376, 
we  obtain 


dt  t/^o    Jy<^    ho         -"  t/xo   Jyo    Izo      {  dt         dz  dt  \ 

^rrr-  %  ^^ +rrr-  ~  dy.    ca) 

'   t/xo     /z/0     Izo         dt  '^0    ^V^     ha  dt  ^ 

pyi    PZi 

In  equation  (2),  Art.  376,  change  u  into  /      /     udzdy. 
Then  finding  the  values  of 


/      /     udz  dy    and     -—  /      /     u  dz  dy. 


d 

~dtdyo  f^zo   'Y  dx 


which  may  be  done  from  (10)  by  changing  x  to  y  and  y  to  z, 
we  have 


METHOD    OF  PARAMETERS.  $21 


It 


Ixq    dy^    Jzo  -^         /xo    Uyo    fJzo       \   dt         dx   dt    \ 

/^i   rvi  /^i     \dz    .    dz  dx  ^   J 

'  ho   /vo  Jzo       \dt^  dx  dt  S  ^. 

u,  and 
employ  equation  (2),  Art.  376,  with  y  put  for  x.     Then  in  the 

resulting  equation  change  u  into  /  u  dz,  and  reduce  by  equa- 
tion (i),  Art.  375,  with  z  put  for  x.     Then  we  obtain 

dt^^^   ho  t/zo  Uxo   Ivo  dzo     y  dt       dy  dt  \ 

'  vfxo    /yo    ho       \dt        dydt  )  '^0    'Vo   e/^0        dt  ^    ^ 

Tn  equation  (A),  Art.  377,  first  change  u  mtoj     tidy,  and 

reduce  the  first  member  by  equation  (i),  Art.  375,  with  x  put 

Zzi 
tc,  reducing  the    first 

member  by  equation  (2),  Art.  376,  with  x  put  for  /,  and  z  for 
X ;  and  lastly,  change  u  into  ut,  performing  the  differentiation 
in  the  first  member.     Then  transposing,  we  have 

£^r/\^ayd.=rr'/\tdy-r'rrutfd. 

t/Xo   t/yo    izo        dx  '^0    *^^o    '^0  "^        dxo    lyo    ho         dx 

Jxo  Jyo   ho     \  \dx  ^  dz  dxi     ^     dzdx  ) 


.522  CALCULUS  OF    VARIATIONS.  I 

Next,  in  equation  (A),  Art.  377,  put 7  for  x^  thus: 
/      -—  dy  =  /     u. 

In  this  equation  first  change  u  into  /    u,  reducing  the  first 

member  by  equation  (2),  Art.  376,  with  y  put  for  t,  and  z  for  x] 
then  change  it  into  Mt,  perform  the  indicated  differentiations 
in  the  first  member,  and  finally  prefix  an  integral  sign  fol- 
lowed by  dx  to  both  members,  as  we  evidently  have  a  right 
to  do.     Then  transposing,  we  have 

/      /      /    u  -—  dy  dx  ^=  I      I  iitdx 

In  equation  (A),  Art.  377,  change  u  into^  u,  and  reduce  the 
first  member  by  equation  (2),  Art,  376,  with  x  put  for  /,  and 
y  for  X.      Then  change  u  into  /     u  dz,  and  reduce  the  first 

1/  Zq 

member  by  equation  (i).  Art.  375.  Lastly,  changing  u  into 
uty  and  transposing,  we  obtain 

/      /      /     u---dzdx-=. 
ixo   lyo  tJzQ  t/xQ   lyo   izq        [  dx       dy  dx  ) 

fJxo   hjo  fJzo     {  \dx    '   dy  dx)      ^       dy  dx  )  ^   ^ 


METHOD   OF  PARAMETERS.  523 

465.   Suppose  now  that  we  had,  as  we  shall  presently  have, 
occasion  to  consider  such  expressions  as 

u  =  r^  r^  r^  vdydx,    u  =  n  r^^  r^  vdz  dx, 


which  we  may  call  mixed  expressions,  V  denoting  any  func- 
tion of  the  independent  variables  x,  y  and  z,  and  of  any  quan- 
tities dependent  upon  them.  Then  it  is  evident  that  equation 
(9)  must  still  hold  for  SU\  and  therefore,  by  putting  V  for  u 
in  formulae  (A),  (B)  and  (C),  we  obtain  at  once  the  equations 

OXq    lyo    tJzQ      [  dy  ) 


524  CALCULUS  OF    VARIATLONS. 

^CCi:  I "" + "I  «^  t  - +/.7.T  "'^-  (■) 

where  ^x,  dy  and  ^z  are  synonymous  with  Dx^  Dy  and  i7^, 
and  (J  F  is  obtained  upon  the  supposition  that  ;ir,  jj/ and  5"  do 

not  vary,  while  -7—,  —r-  and  -—-  are  the  partial  differential  co- 

-^  ax     dy  dz 

efficients  of  V  taken  only  under  the  supposition  that  x,  y  and 
z  enter  V  explicitly. 

Moreover,  if  in  any  of  the  formulae  {A),  {B),  (Q,  {D),  etc., 
we  wish  to  substitute  in  the  first  member  but  one  limit  of  the 
variable,  we  need  merely  substitute  in  the  second  member  the 
same  single  for  the  double  limits  of  that  particular  variable, 
none  of  the  other  substitutions  being  in  any  way  affected. 


Problem  LXVII. 

456,  //  is  required  to  determine  the  form  which  must  be 
assumed  by  a  surface  of  given  area  in  order  that  it  may  enclose  a 
m,aximum-  volurne. 

This  is  only  the  most  general  statement  of  the  problem, 
particular  cases  of  which  were  discussed  in  Prob.  LXIV.  and 
also  in  Prob.  XVI.  Denote  the  volume  by  v.  Then,  although 
we  might  express  vhy  a.  double  integral,  we  shall,  for  greater 
generality,  write 

v=    I      I      fdzdydx.  (i) 

t/ao   *^yo  t/«o 


Now  the  surface  6"  which  bounds  this  volume,  and  which 
is  to  retain  an  invariable  area,  must  be  supposed,  as  usual  in 
triple  integrals,  to  consist  of  the  six  hmiting  faces  C^,  C^,  B^,  B^, 
A^  and  A^,  S  being  their  sum.  Moreover,  these  several  faces 
are  expressed  by  the  equations 


(2) 


SOLID  OF  MAXIMUM  VOLUME.  525 

/^o     /'yi      /»^1 

1^1  rvi  nzi 

^.=/  JyJ.,d^dy' 

where  /  —  ~,  /  =  — 1  and  q  =  — -.     But  assuming-  —  ^  as  the 
ax  dx  dy  ^ 

constant  multiplier,  we  are,  by  the  method  of  Euler,  to  maxi- 
mize absolutely  the  expression 

U=v-a{A,-\-A^-\-B,-\-B,^C,-\-C:)=v-aS.       (3) 

Here  the  limits  of  the  integrals  are  subject  to  no  explicit 
restrictions,  and  it  might  therefore  appear  as  though  no  maxi- 
mum could  be  possible  ;  but  problems  of  relative  maxima  and 
minima  do  not  always  require  any  additional  explicit  restric- 
tions upon  the  limits,  the  implicit  restriction  that  the  variations 
should  be  so  taken  as  to  render  one  or  more  of  the  integrals 
constant  being  sufficient  for  a  definite  solution.  In  the  pres- 
ent case  it  will  be  found  that  the  fact  that  5  is  to  maintain 
always  the  same  value  constitutes  an  implicit  restriction, 
which  is  sufficient. 

457.  From  formulae  (H),  (I)  and  (G),  by  putting  for  V 
I  in  the  first,  \^\  -\-  y""  in  the  second,  and  '/i+/''-j-^'  in  the 
third,  and  substituting  but  one  limit,  say  the  inferior,  in  the 
first  member,  we  obtain 


526  CALCULUS  OF  VARIATIONS. 

f*Xy       lyo        PZ^  y' 

dB,=  /      /         ^    ^      ^dy'dzdx 

+£y^C\   VT^^^6,+  VT^{^^qdy^dx 

l^i    IVo     Pz-i      ^ _    ^ 


A Sq\.dy  dx 


/x\  f*yi  izq    , 


(4) 


and  similar  equations  with  the  single  suffixes  changed  from 
o  to  I  will  evidently  hold  for  SA^,  dB^  and  SC^. 

We  now  transform  by  equations  (D),  (E)  and  (F)  the  terms 
containing  dp,.dq  and  dy\  which  may  be  done  by  substituting 
in  these  equations,  respectively, 

_  P  dt  _  dSz  _  r. 

^  ~~   vTIfT^^-p^^'        'dx~  dx  ~    ^' 

a  dt       dSz        ^ 


II  = 


|/i  4-y^'  dx        dx 


observing  that 


dt       dSz  J      dt       dSy 

~         =zo         and     --  =  -f-  =  o, 
dz       dz  dy        dy 


SOLID   OF  MAXIMUM  VOLUME.  52; 


aud  also  that 


du  _,du     _  du  du  _.du      _  du 

dx       dz  dx'  dy       ds  dy^ 

and 

du  j^du  ,  _du 

dx       dy  dx^ 

where  the  differentials  in  the  second  members  are  merely 
total.  Having  effected  these  transformations,  and  taken  also 
the  variations  oi  vm  (i),  we  shall  have 

6v  =  /     /    6zdydx-\- I     /     /     dy  dz  dx 

l^x  f*y^   r^x 
+  /     /      /     dxdzdy, 

/xo   j'*yi  /zi  /xo   /yi   pzi 

6B^^-rrr±-j=-6yd.d. 

Jxo     I        e/zo    dx    4/1  -\-y     ^ 

+ -  \Szdydx 

^  dy   y'i_|_/+^«^ 

/>..  /../.of       q-py'       ^^       Vi^f^q'Sy  I  dx 

/^i   f*y\  1^0  c  4)  , ) 

+  /      /     /     )— -  ^  6z+Vi+p''+q''dx[dy, 


(5) 


528  CALCULUS  OF  VARIATLONS. 

and  it  is  evident  that  SA^,  SB^  and  SC^  will  be  expressed 
by  precisely  the  same  equations  with  every  single  suffix  o 
changed  to  i. 

458.  But,  as  appears  from  (3),  we  must,  to  obtain  dU, 
multiply  the  last  three,  or  rather  the  last  six,  of  equations  (5) 
by  —  a,  and  add  the  result  to  Sv.  If,  then,  in  this  equation  we 
resolve  all  the  signs  of  substitution,  and  then  bring  together 
the  terms  which  contain  like  variations,  and  are  affected  by 
like  substitutions  (which  M.  Sarrus  does),  SU  will  consist  of 
thirty  distinct  terms,  six  holding  throughout  the  six  limiting 
faces,  and  the  remaining  twenty-four  referring  to  what  might 
be  termed  the  twenty-four  edges  of  these  six  faces,  each  actual 
edge  of  the  body  being  regarded  as  belonging  to  either  of  two 
faces  ;  and  these  thirty  terms  are  independent. 

But  following  the  device  of  Moigno  and  Lindelof,  we  may 
write  SUin  the  following  condensed  manner: 


t/o-o    t/2/0      Izo      j  \dx    |/i 


ap 


+  /+/ 


d 


dy    ^-^j^p'^j^q 


)\ 


"'^         \\dsdydx 


-^T'  T'  T'dxdzdy 


SOLID   OF  MAXIMUM   VOLUME,  $2g 

+(/±   VT+ynSx  id^:^0,  (6) 

where  the  signs  of  substitution  denote  throughout  SW  the 
same  series  of  operations  as  before ;  the  sign  ±  in  the  first 
three  terms  denoting  that  at  the  first  substitution  the  upper, 
and  at  the  second  the  lower  is  to  be  taken ;  the  same  sign  in 
the  last  three  terms  signifying  that  the  upper  is  to  be  taken 
when  the  quantities  substituted  have  the  same  suffix,  and  the 
lower  when  they  have  not,  while  these  results  must  still,  as 
the  sign  —  or  +  indicates,  be  multiplied  by  —  i  or  +  i>  ac- 
cording as  the  quantity  above  the  left-hand  sign  of  substitution 
has  the  suffix  i  or  o.  But  the  reader  who  may  prefer  can 
easily  write  out  the  thirty  terms  from  (5),  and  verify  directly 
the  last  and  the  following  assertions. 

459.   Now  equating  to  zero  the  coefficient  of  d^  in  the 
first  term  of  (6),  we  have 

Or  denoting  by  r  and  r'  the  principal  radii  of  curvature,  the 
last  equation  is  equivalent  to 

i  +  l=i        and    !  +  -,=  --;  (7) 

r       r        a  r       r  a 


530  CALCULUS   OF  VARLATIONS. 

the  first  holding  throughout  the  face  C^,  and  the  second 
throughout  the  face  C^.  Then  the  two  equations  (7)  show 
that  C^  and  C^  have  their  mean  curvatures  constant,  but  turn 
their  convexities  in  opposite  directions. 

Equating  to  zero  the  coefficients  of  dz  in  the  fourth  term, 
we  have 

'-^''  =Ti,  (8) 


ni+/+/)(i+/o 

which  involves  the  four  equations  relative  to  the  intersection 
of  the  (7's  and  ^'s,  the  negative  sign  holding  for  the  edges 
C^  B^  and  C^  B^,  and  the  positive  sign  for  the  edges  C^  B^  and 
C^B^,  But  the  first  member  of  (8)  equals  the  cosine  of  the 
angle  made  with  each  other  by  the  two  surfaces  along  their 
common  intersection ;  and  since  this  cosine  is  unity,  we  infer 
that  the  ^'s  and  Cs  are  always  tangent  or  accord  along  their 
common  edges. 

We  observe,  also,  that  equation  (8)  will  Cause  the  coeffi- 
cient of  (^K  in  the  same  fourth  term  to  vanish  without  giving 
any  additional  equations.  For  since  the  ^'s  and  ^^'s  are 
tangent,  /  and  q  will  have  the  same  meaning  in  both  along 
their  intersection  :  thus  ten  equations  have  been  considered. 

Equating  to  zero  the  coefficient  of  ^z  in  the  fifth  term  of 
(6),  we  have,  for  the  intersections  of  the  ^'s  and  (7's, 

^         =Ti;  (9) 


4/1+/+^' 


—  I  when  the  suffixes  are  alike,  and  +  i  when  they  are  un- 
like. Equation  (9)  denotes  that  the  faces  C^  and  C^  must,  along 
their  intersections  with  the  planes  A^  and  A^^  be  normal  to  the 
axis  of  X ;  that  is,  they  must  be  tangent  to  or  accord  with  these 
planes. 

TJien  we  observe,  as  before,  that  (9)  causes  the  coefficient 
of  Sx  in  this  same  fifth  term  to  vanish  without  giving  rise  to 


SOLID   OF  MAXIMUM    VOLUME.  53 1 

any  additional  equations.     Thus,  then,  eight  terms  more  have 
been  caused  to  disappear. 

460.  Now  having  equated  to  zero  the  second  term  in  (6), 
which  is  relative  to  the  cylinders  B^  and  B^,  and  remembering 
that  dy  must  remain  constant  along  any  particular  generatrix, 
but  is  independent  for  each,  we  shall  obtain 


i 


^^1  V  n        ay 

^  dz  =  o,  (loj 


dx  i/i-\-y 


which,  with  the  positive  sign,  holds  for  any  generatrix  of  B^, 
and  with  the  negative  sign  for  the  corresponding  generatrix 
of  B^.  But  as  the  integration  in  (lo)  is  to  be  effected  regard- 
ing X,  y  and  y'  as  constant,  we  have 

^•^         -..-..)  =  0.  (II) 


dx  |/i  -|-y 


Equating  the  first  factor  to  zero,  we  would  obtain  a  cylinder 
of  radius  a,  the  limits  of  z  being  wholly  undetermmed.  But 
neglecting  for  the  present  this  supposition,  we  must  have 
z^  =  Zj]  that  is,  B^  and  B^  vanish  or  reduce  to  mere  edges. 

The  condition  z„  =  z^  will  cause  also  the  first  member  in 
the  last  term  of  (6)  to  vanish  without  giving  any  new  equa- 
tions ;  so  that  thus  six  more  terms  in  all  disappear. 

Equating  to  zero  the  third  term  in  (6),  and  remembering 
that  Sx^  and  dx^  are 'two  independent  constants,  we  have 

r^  r'dzdy  =  o,  (i2) 

an  equation  which  involves  two,  as  it  holds  for  either  of  the 
faces  A^  or  A^,  and  shows  at  once  that  these  two  plane  faces 
must  also  disappear. 

Then  (12)  will  cause  the  last  member  in  the  last  term  of 
(6),  which  is  relative  to  the  four  intersections  of  them's  and 


532  CALCULUS  OF  VARIATLONS. 

B'%  to  vanish  without  giving  any  new  equations.      Thus  all 
the  terms  in  (6)  have  been  caused  to  vanish  severally. 

461.  If  we  admit  into  the  solution  the  cylinder  with  ra- 
dius a^  or  for  A^  and  A^  any  edge  perpendicular  to  the  axis  of 
X  (which  is  probably  admissible),  we  cannot  say  that  all  the 
conditions  of  the  question  could  be  satisfied.  But  if  we  as- 
sume B^  and  B^  to  become  mere  edges,  and  A^  and  A^  to 
become  points  only,  the  volume  in  question  must  be  entirely 
enclosed  by  the  curved  faces  C^  and  C^, 

Moreover,  from  what  has  been  shown  it  will  appear  that 
these  two  faces  must  be  respectively  perpendicular  to  the  plane 
of  xy  along  their  common  intersection,  and  they  must  there- 
fore meet  in  such  a  manner  as  to  coalesce  and  to  form  one  and 
the  same  surface,  which  will  be  given  by  the  equation  derived 
from  (7), 

l-+a=y  (■3) 

,  Now  the  sphere  of  radius  2a  will  evidently  satisfy  all  these 
conditions.  But  in  order  to  exclude  all  other  hypotheses,  it 
would  still  be  necessary  to  show  that  the  sphere  is  the  only 
admissible  solution  obtainable  by  equating  to  zero  the  terms 
of  the  first  order  in  SU.  But  the  proof  of  this  fact  has  never 
yet  been  obtained  by  analyses ;  and  even  if  it  could  be,  it  would 
still  be  necessary  to  show  that  the  sphere  would  cause  the 
terms  of  the  second  order  to  become  always  positive,  or  else 
those  of  some  other  even  order  to  become  so,  the  preceding 
having  reduced  to  zero ;  and  this  would  present  a  new  and 
probably  an  insurmountable  difficulty.  Moreover,  as  we  take 
the  entire  sphere,  we  shall  be  obhged  to  deal  with  some  quan- 
tities which  will  become  infinite ;  which  fact  might  of  itself 
throw  some  doubt  upon  our  investigations.  But  although 
the  complete  discussion  of  this  problem  appears  to  be  beyond 
the  power  of  the  present  methods  of  analysis,  we  are  assured 
from  other  considerations  that  the  sphere  is  its  true  and  only 
solution. 


CHAPTER   IV. 

APPLICATION   OF  THE    CALCULUS    OF   VARIATIONS   TO   DETER- 
MINING   THE   CONDITIONS    WHICH    WILL   RENDER  A 
FUNCTION'  INTEGRABLE    ONE   OR   MORE  TIMES. 


Section  I. 

CASE  IN    WHICH   THERE  IS  BUT  ONE    INDEPENDENT 
VARIABLE. 

Problem  LXVIII.  i 

462.  Suppose  we  seek  by  the  calculus  of  variations  to  maxi- 
mize or  minimize  the  expression 

Then  returning  to  our  former  notation,  we  shall  have 

dx  y      /       y 

d'Q-n"-  _^    ^y"  I  ^^y\ 


534  CALCULUS  OF   VARLATIONS. 

SO  that  the  equation 

M=N-P'-\-Q"z=:0 

will  reduce  to  o  =  o ;  that  is,  M  will  vanish  of  itself,  or  identi- 
cally ;  so  that  we  obtain  no  equation  from  which  we  can 
derive  a  general  solution,  and  have  left  merely  the  terms  at 
the  limits,  which  may  be  written 

4-63.  Now  in  seeking  to  explain  this  anomaly,  we  observe 
that  Fmay  be  written 

y{xy"  -\ry')-xyy' 

f 
Whence  we  see  that 

fvd.  =  ^-l+c,        and     U  =  iy-l.  (3) 

Thus  it  appears  that  U  can  in  this  case  be  freed  from  the 
sign  of  integration,  and  that  the  discussion  of  the  conditions 
which  will  render  it  a  maximum  or  a  minimum  does  not, 
strictly  speaking,  belong  to  the  calculus  of  variations ;  and  we 
can  readily  show  that  whenever    U  is  integrable,  M  must 

vanish  identically.      For  assume  the  equation   U—J^     Vdx, 

where  V  is  any  function  of  x,  y,  y\  ^  -^  .  .  y^'^\  but  is  of  such 
a  form  that  Vdx  shall  be  immediately  integrable ;  that  is,  in- 
tegrable independently  of  any  relations  which  may  hold  be- 
tween X  and  y.  Then  we  know  that  by  definite  integration 
U  may  be  written 


^-l> 


_   x,y,y,.,..y(^-'^)),  (4) 


CONDITIONS  OF  INTEGRABILITY.  535 

which  shows  that  U  depends  solely  upon  the  limiting  values 
of  X,  y,  y,  etc.,  the  relations  between  x  and  y  being  altogether 
in  our  power.  Now  if  in  U^  before  integration,  we  vary  j^,j/', 
etc.,  but  suppose  these  quantities  at  the  limits  to  remain  fixed, 
Z7will  undergo  no  change;  that  is,  SU  will  vanish;  and  be- 
cause ^y,  dy\  etc.,  are  zero  at  the  limits,  ^^can,  as  usual,  be 
reduced  to  the  form 

SU=£-MSydx^o,  (5) 

to  satisfy  which,  since  Sy  is  in  our  power,  M  must  vanish. 
But  unless  M  vanish  identically,  we  shall,  by  equating  it  to 
zero,  have  an  equation  which,  if  it  be  integrable,  will  deter- 
mine J/  as  a  function  of  x,  or,  if  not  integrable,  will  establish 
an  implicit  or  differential  relation  between  them,  both  of  which 
are  contrary  to  the  conditions  of  the  question.  If,  therefore, 
Vdx  be  integrable  immediately — that  is,  without  assuming 
any  particular  relation,  either  explicit  or  implicit,  between  x 
and  y—M  must  vanish  identically. 

464.  Conversely,  if,  Z7and  F  having  the  same  meaning  as 
before,  we  fin'd  M  to  vanish  identically,  we  may  conclude  that 
Vdx  is  immediately  integrable.  For  we  see  that  SU  will  in 
this  case  consist  of  the  terms  at  the  Hmits  only,  as  in  equation 
(2),  so  that  we  infer  that  C/must  depend  solely  upon  the  values 
which  X,  y^  y' ,  etc.,  may  have  at  the  limits  ;  and  hence  that  U 
must  in  reaUty  be  a  function  of  these  quantities  only,  which, 
so  long  as  y  is  wholly  in  our  power,  could  not  be  true  unless 
Vdx  were  immediately  integrable. 

This  mode  of  reasoning  would  seem  to  be  sufficiently  con- 
clusive ;  nevertheless  it  is  not  so  regarded  by  Prof.  Tod- 
hunter,  and  the  reader  will  find  in  his  Integral  Calculus,  Art. 
382,  an  attempt  at  a  more  rigorous  demonstration. 


53^  CALCULUS  OF   VARIATIONS. 


Problem  LXIX. 

465.  V  having  the  same  form  as  before^  it  is  required  to  de- 
termine the  conditiofis  which  will  render  V  immediately  integrable 
any  num,ber  of  times,  m. 

First  assume  vt  to  be  2,  and  we  have 

f  I  fvdx  \  dx  =  xjvdx  -Jx  Vdx  ;  (i) 

and  hence,  to  insure  that  F  shall  be  twice  immediately  inte- 
grable, we  must  have  both  F'and  Vx  immediately  integrable; 
and  conversely,  if  these  quantities  be  immediately  integrable 
once,  Fwill  be  immediately  twice. 
Now  the  first  condition  will  give 

N-  P'-\-  Q"  -  etc.  =  J/=  o,  (2) 

which  must  be  true  identically ;  while  putting  v  for  Vx,  the 
second  condition  will  give,  in  like  manner, 

Vy  —    Vy/-\-  Vyn  "—     CtC.    ==    O,  ^  (3) 

which  must  also  be  true  identically.     But  (3)  may  be  replaced 
by  another  equation,  thus  : 

Vy—X    Vy,  VyI  =   X    Vyf ,  Vyt,  =    X    Vy>f  ,     CtC, 

Vy/=    X    Vy/+    Vyr,  Vy.' ^   =   X    Vy.' '  +   2  Vy.\ 

Vy./"  =  xVy,.'"+2>Vy./',       etc. 

Substituting  these  values  in  (3),  and   omitting    those  terms 
which  are  known  to  be  zero  by  (2),  we  shall  obtain 

P-2Q-\-zR"-Qic.  =  o,  (4) 

which  must  be  true  identically. 


CONDITIONS  OF  INTEGRABILITY.  53/ 

Hence  that  Fmay  be  immediately  integrable  twice,  equa- 
tions (2)  and  (4)  must  be  identically  true. 

Now,  in  the  more  general  case  in  which  fn  is  any  number 
less  than  n^  it  is  generally  shown  in  works  on  the  integral 
calculus  that,  if  we  denote  by  U  the  result  of  the  integration 
oi  V  m  times,  we  may  exhibit  (7  thus : 

U= — ^ — \  ^"^-1  fvdx  —  (m—  i)x-^-^  fxVdx 


1 

{in  —  i){m  —  2) 

1-2 


x^-^fx''Vdx-ttc.±fx^-^Vdx\.    (5) 

Whence  it  appears  that  to  render  F integrable  //^  times  it  is 
necessary  and  sufficient  that  the  quantities  V,  Vx,  Vx"",  .... 
y^m-i  shall  be  severally  integrable  ;  and  the  equations  arising 
from  these  conditions  can  be  determined  precisely  as  before. 
Thus  if  m  be  3,  we  shall  find,  in  addition  to  equations  (2)  and 
(4),  the  identical  equation 

e-^i?'  +  i^5''-etc.  =  a  (6) 

1-2  1-2  ^  ^ 


Problem  LXX. 

466.  //  is  required  to  determine  the  conditions  which  will 
render  Vdx  immediately  integrable,  V  being  any  function  of  x 
and  the  dependent  variables  y  and  z,  together  with  their  differen- 
tial coefficients  with  respect  to  x  ;  that  is,  y\  y" ,  z' ,  z" ,  etc. 

Putting,  as  before,  U  for  the  integral,  and  transforming 
SU,  we  shall  obtain,  as  in  Art.  303,  a  result  which  may  be 
written 

SU^L,-L,^  r^M dydx  +  r^YSz dx,  (i) 

where 

M=  Vy-  Vy/+  Vy>/'-  ctc,       N  =  V, -  F/+  F,//'-  etc.  (2) 


538  CALCULUS  OF    VARLATIONS. 

Now,  as  before,  we  may  suppose  the  limiting  values  of  j/,  y, 
z,  z',  etc.,  to  be  fixed,  so  that  L^  and  L^  will  vanish. 

Moreover,  Sy  and  Sz  are  entirely  independent,  so  that  M 
and  TV  must  severally  vanish  if  C/is  to  depend  solely  upon  the 
limiting  values  of  x,  y,  y',  z,  z',  etc.  But  either  or  both  the 
equations  M  —o  and  N  —  o,  unless  they  be  identically  true, 
would  enable  us  to  establish  some  explicit  or  implicit  rela- 
tion between  x,  y  and  z,  whereas  we  require  that  Vdx  shall 
be  integrable  irrespectively  of  any  such  relation,  other  than 
that  jj/  and  z  are  to  be  regarded  as  functions  oi  y  and  x. 

If  V  were  integrable  m  times,  it  is  easy  to  see  that,  as  in 
Prob.  LXIX.,  we  must  have  F,  Vx,  Vx\  etc.,  immediately 
integrable,  since  equation  (5)  of  that  problem  requires  merely 
that  V  shall  be  a  function  of  x,  and  it  might,  therefore,  con- 
tain any  number  of  dependent  variables,  y,  z,  ti,  and  their  dif- 
ferential coefficients  with  respect  to  x.  Hence  we  should 
evidently  obtain  with  such  equations  as  (2),  (4)  and  (6)  similar 
equations  in  z. 

Moreover,  it  will  appear  that  for  any  other  dependent 
variable  ti  which  V  may  contain,  we  shall  require  in  addition 
a  similar  set  of  equations  in  u. 


Section  II. 

CASE  IN  WHICH  THERE  ARE  TWO  INDEPENDENT  VARIABLES. 

Problem  LXXI. 

467.  It  is  required  to  determine  the  conditions  ivJiich  will 
render  I    I  Vdy  dx  reducible  to  a  single  integral,  ivhere  V  is  any 

function  of  x,  y,  z,  p  and  q,  x  and  y  being  tzvo  independent  vari- 
ables,  and  p  and  q  partial  differentials  of  z  with  respect  to  these 
variables. 

Denoting  the  definite  integral  by  U,  we  know  that  after 
transformation  (JC/may  be  written 


CONDITIONS   OF  INTEGRABILITY.  539 

dU=L  +fJ'fJ'M  Ss  dydx,  (i) 

where  Z,  although  consisting  of  simple  integrals,  involves  only 
terms  which  relate  to  the  limits  of  the  integration ;  and  by 
supposing  z  to  be  unvaried  along  the  lines  x  —  x^  and  x  =  x^, 
we  can  make  L  consist  only  of  quantities  which  are  functions 
of  X,  and  the  variations  of  such  quantities. 

Now  we  know  that  if  we  regard  all  the  quantities  at  the 
limits  as  fixed,  L  will  vanish,  so  that  if  U  can  be  reduced  to  a 
single  integral  depending  upon  these  quantities  only,  31  must 
vanish ;  and  if  this  reduction  is  to  be  possible  without  deter- 
mining z  as  some  function,  explicit  or  imphcit,  of  x  and  y,  M 
must  vanish  identically,  otherwise  the  equation  M  —  o  will 
establish  some  such  relation. 

On  the  other  hand,  if  J/ vanish  identically,  (^^  will  reduce 
to  Z,  and  we  infer  that,  as  it  depends  solely  upon  quantities  at 
the  limits,  U  is  immediately  reducible  to  a  single  integral. 

468.  Now  we  can  determine  what  form  Fmust  have  to 
render  this  reduction  possible.     For 

M=^V,-  F/- F,,;  (2) 

and  if  M  is  to  vanish  identically,  it  is  evident,  in  the  first  place, 
•  that  all  terms  containing/,  q,  r,  j and  /  must  vanish.  But  we 
have  seen  (Art.  428)  that  when,  and  only  when,  Fis  of  the 
form/j  +/2/  4-/3^,  the  equation  M  —o  will  fail  to  rise  above 
an  order  n  —  2\  that  is,  2  —  2. 

Such,  then,  must  be  the  form  of  F;  but  that  J/ may  en- 
tirely vanish  it  will  be  necessary,  in  addition  to  this,  that/j, 
f^  and  /g,  which  are  all  functions  of  x,  y  and  z  only,  shall  be 
subject  to  a  certain  additional  relation.     For  we  have 

V^=j-^'        ^P^f^'        v^=U 


540  CALCULUS  OF  VARIATIONS. 

SO  that  the  equation  M  =  o,  after  rejecting  terms  containing 
differential  coefficients  of  z,  gives 

dz       dx        dy         ^ 

and  the  /'s  must  be  so  related  as  to  render  this  equation  also 
identically  true. 

4-69.  Although  not  very  rigorously  demonstrated,  the 
foregoing  are  all  the  leading  theorems  relative  to  this  subject, 
and  it  would  be  unprofitable  to  pursue  it  further.  For  while 
the  calculus  of  variations  gives  us  the  means  of  determining 
whether  or  not  V  is  immediately  integrable,  it  does  not  of 
itself  indicate  the  method  of  effecting  the  integration;  and 
this  method  is  what  we  wish  chiefly  to  know. 

The  theorems  given  in  the  preceding  problems  relative  to 
this  subject,  which  is  often  called  the  theory  of  integr ability,  or 
the  conditions  of  integr ability,  can  be  established  without  the 
aid  of  the  calculus  of  variations,  but  less  easily. 

The  reader  who  may  wish  to  pursue  this  subject  further 
is  referred  to  the  treatise  on  the  calculus  of  variations  by  Prof. 
Jellett,  Chap.  X.,  and  also  to  Todhunter's  History  of  the  Calc. 
of  Van,  Chap.  XVII. 


CHAPTER   V. 

HISTORICAL     SKETCH   OF    THE     RISE  AND     PROGRESS    OF  THE 
CALCULUS   OF  VARIATIONS. 

470.  Questions  of  maxima  and  minima  were  among  the 
first  to  occupy  the  attention  of  mathematicians  after  the  in- 
vention of  the  differential  or  fluctionary  calculus,  which, 
according  to  Woodhouse,  occurred  about  the  year  1684,  or 
three  years  prior  to  the  publication  of  the  Principia.  The 
ordinary  calculus  was  not,  however,  given  to  the  world  at 
once  in  a  single  treatise,  but  was  developed  gradually  in 
essays,  in  communications  to  learned  societies  and  journals, 
and  in  letters  between  men  of  science. 

The  first  question  considered,  of  that  particular  species  of 
maxima  and  minima  which  forms  the  chief  subject  of  the  cal- 
culus of  variations,  appears  to  have  been  that  of  the  solid  of 
minimum  resistance ;  and  this  was  first  proposed  by  Newton 
in  the  Principia.  But  although  Newton  was  the  first  to  con- 
sider a  question  belonging  to  the  calculus  of  variations,  no 
importance  seems  to  have  been  attached  to  this  problem  either 
by  himself  or  his  contemporaries,  and  it  did  not  become  at  that 
time  the  subject  of  discussion. 

The  true  beginning  of  our  science  dates  from  the  month 
of  June,  1696,  when  John  Bernoulli,  Professor  of  Mathema- 
tics at  Groningen,  proposed  in  the  Acta  Eruditorum  (or 
Doings  of  the  Learned),  a  work  then  published  at  Leipsic, 
and  at  that  time  the  chief  medium  of  communication  between 
men  of  science  and  letters,  the  following  problem  : 


542  calculus  of  variations. 

"Problema  Novum* 
"Ad  cujus  solutionem  mathematici  invitantur. 
"  Datis  in  piano  verticali  duobus  punctis  A  et  B,  assignare 
mobili  J/viam  AMB,  per  quam  gravitate  sua  descendens,  et 
moveri  incipiens  a  puncto  A,  brevissimo  tempore  perveniat 
ad  alterum  punctum  B.'' 

This  problem  engaged  at  once  the  attention  of  Leibnitz, 
James  Bernoulli,  brother  of  John,  and  Professor  of  Mathe- 
matics at  Basle,  and  the  Marquis  de  I'Hopital,  the  first  two 
of  whom  appear  to  have  solved  the  problem  within  the  allot- 
ted time,  which  was  six  months.  Leibnitz  at  once  forwarded 
his  solution  to  the  proposer,  asking  that  it  might  not  be  imme- 
diately published,  in  order  that  other  mathematicians  might 
be  encouraged  to  attempt  the  problem  ;  and  he  subsequently, 
as  no  solution  appeared,  requested  that  the  period  might  be 
extended,  a  request  with  which  John  Bernoulli  compKed,  and 
accordingly  reannounced  the  problem  in  a  programme  dated 
at  Groningen,  Janua;-y,  1697.  Upon  learning  of  this  exten- 
sion, James  Bernoulli  retained  his  solution,  being  desirous,  as 
he  stated,  of  investigating  and  adding  to  the  problem  certain 
others  of  a  similar  character ;  which  he  did,  as  we  shall  subse- 
quently see. 

In  the  Acta  for  the  following  May  were  published  the  solu- 
tions of  the  two  Bernoullis,  together  with  one  by  De  I'Hopi- 
tal, the  last  being  without  demonstration.  James  is  in  advance 
of  his  brother ;  but  as  his  solution  is  given  by  Woodhouse,  it 
will  here  suffice  to  say  that  both  brothers  assume  the  prin- 
ciple that  whatever  maximum  or  minimum  property  is  pos- 
sessed by  the  whole  of  any  required  curve  must  be  possessed 
also  by  every  portion  of  the  curve ;  and  that  therefore,  if 

*  "  A  New  Problem,  to  the  solution  of  which  mathematicians  are  invited. 

"  Given  two  points  A  and  ^  in  a  vertical  plane,  to  find  for  the  movable  (par- 
ticle) M,  the  path  A  MB,  descending  along  which  by  its  own  gravity,  and  begin- 
ning to  be  urged  from  the  point  A,  it  may  in  the  shortest  time  reach  the  other 
point  B.'' 


HISTORICAL    SKETCH.  543 

we  consider  the  required  curve  as  a  polygon  of  an  infinite 
number  of  sides,  it  will  be  sufficient  to  consider  two  consecu- 
tive sides  or  elements.  But  this  principle,  while  it  enabled 
them  to  obtain  in  this  case  a  correct  result,  is  not  universally 
true. 

47 (.  At  the  close  of  his  paper  James  Bernoulli  proposed 
two  additional  problems :  first,  to  determine  the  curve  of 
quickest  descent  from  a  given  fixed  point  to  a  given  vertical 
line ;  and  second,  among  all  curves  having  a  given  length  and 
a  given  base,  to  find  the  curve  such  that  a  second  curve,  each 
of  whose  ordinates  is  some  function  of  the  corresponding  or- 
dinate or  arc  of  the  first,  may  contain  a  maximum  or  minimum 
area.  But  although  in  the  first  of  these  problems  we  have 
a  particular  case  of  the  question  subsequently  considered  by 
Lagrange  as  to  what  conditions  must  hold  at  the  limits  in 
maximizing  or  minimizing  a  definite  integral,  little  appears 
to  have  been  effected  in  this  direction  prior  to  the  re- 
searches of  that  mathematician ;  so  that  we  shall  follow  the 
second  problem  only. 

The  second  case  of  this  problem  led  to  an  acrimonious 
discussion  between  the  Bernoullis  which  was  little  creditable 
to  John.  For  still  adhering  to  the  principle  mentioned  at 
the  close  of  the  last  article,  which  in  this  case  fails,  owing  to 
the  isoperi metrical  property  that  the  required  curve  must 
have  a  given  length,  he  continually  obtained  erroneous  re- 
sults ;  nor  would  he  frankly  acknowledge  his  error  after  it 
had  been  indicated  by  his  brother.  No  solution  of  this  prob- 
lem, however,  appeared  until  James  BernouUi,  in  May,  1701, 
published  his  in  the  Acta;  although  a  solution  without  demon- 
stration had  appeared  in  the  Acta  for  the  preceding  June.  In 
this  demonstration  three  consecutive  elements  of  the  required 
curve  are  taken  instead  of  two,  and  a  rude  mode  of  imposing 
the  isoperimetrical  condition  is  shown.  A  solution  much 
more   finished,   but   evidently    borrowed    from   that   of    his 


544  CALCULUS  OF  VARLATIONS. 

brother,  was  subsequently  published  by  John  Bernoulli  in  the 
Memoirs  of  the  Academy  of  Science  for  1718  ;  and  this  solution 
may  be  found  in  Woodhouse's  Isoperimetrical  Problems. 
But  no  further  advance,  worthy  of  notice  in  a  sketch  like 
the  present,  appears  to  have  been  made  until  the  advent  of 
Euler. 

472.  The  first  contribution  of  this  mathematician  to  our 
science  was  a  memoir  published  in  the  sixth  volume  of  the 
Ancient  Commentaries  of  Petersburg,  1733.  In  this  memoir 
Euler,  taking  up  the  subject  where  it  had  been  left  by  the 
Bernoullis,  divided  his  problems  into  classes  :  the  first  including 
those  of  absolute  maxima  and  minima,  the  second  those  prob- 
lems of  relative  maxima  and  minima  in  which  but  one  restriction 
is  imposed  upon  the  variations,  as  in  the  problem  of  the  bra- 
chistochrone  when  the  path  is  to  have  a  given  length ;  while 
the  third  included  those  relative  problems  in  which  two  re- 
strictions are  imposed,  as  when  the  brachistochrone  path  is 
to  have  a  given  length  and  also  to  enclose,  with  the  aid  of  its 
extreme  ordinates  and  the  axis  of  x,  a  given  area.  The  erro- 
neous principle  that  the  maximum  or  minimum  property  of 
the  whole  curve  belongs  to  each  portion  also  was  virtually 
adopted,  two  consecutive  elements  only  being  considered  in 
the  problems  of  the  first  class,  three  in  those  of  the  second, 
and  four  in  those  of  the  third.  Nevertheless,  as  he  proceeded, 
he  established  and  tabulated  formulse — twenty -four  in  all — for 
the  various  cases  which  might  arise ;  and  by  this  means  was,  at 
the  close  of  this  memoir,  much  in  advance  of  the  Bernoullis. 

About  the  year  1740  or  1741  Euler  summed  up  his  re- 
searches in  a  second  memoir  published  in  the  eighth  volume 
of  the  Commentaries  of  Petersburg,  the  date  of  the  volume 
being  1736.  But  this  date  proves  nothing,  as  the  same  volume 
contains  observations  made  in  1740.  Euler  had  now  discov- 
ered that  when  F  is  a  function  of  x,  y,  y,.  .  .  .  y^^\  his  previous 
formulae  might  be  expressed  by  one  more  general  formula. 


HISTORICAL   SKETCH.  545 

which  is  still  in  use,  and  which  we  have  denoted  by  the  equa- 
tion M  =  o. 

Also,  the  principle  that  the  maximum  or  minimum 
property  of  the  whole  curve  will  belong  equally  to  every 
portion  was  examined,  and  shown,  in  some  cases  at  least,  to 
be  untrue  ;  and  lastly,  some  advance  seems  to  have  been  made 
in  the  treatment  of  problems  of  relative  maxima  and  minima. 
By  this  memoir  the  calculus  of  variation  was  greatly  im- 
proved, and  it  contained,  in  fact,  nearly  all  that  its  author 
ever  discovered  relative  to  this  subject,  although  in  a  very 
confused  and  ill-arranged  form. 

In  1744  Euler  published  a  tract  entitled  Methodiis  inveni- 
endi  Lineas  curvas  Proprietate  maxwii  mininiive  gauderitesJ^ 
This  work,  which  was  the  most  famous  of  its  author  re- 
lative to  this  subject,  displa3^s  an  amount  of  mathematical 
skill  almost  unrivalled.  Problems  were  here,  as  at  present, 
divided  into  two  great  classes,  absolute  and  relative,  and  the 
treatment  of  the  second  was  for  the  first  time  reduced  to  a 
perfect  science  by  the  discovery  of  the  artifice  still  employed, 
and  termed  the  Method  of  Euler.  This  work  is  also  general- 
ly clear  and  systematic,  containing  an  abundance  of  examples, 
including,  with  many  others,  most  of  those  given  by  us  m 
our  first  chapter ;  and  at  its  close  Euler  had  carried  our 
science  so  far  beyond  the  point  which  had  been  reached  by 
the  Bernoullis  that  he  may,  almost  equally  with  Lagrange,  be 
regarded  as  the  author  of  the  calculus  of  variations.f 

Euler  subsequently  published  in  the  tenth  volume  of  the 
New  Commentaries  of  Petersburg,  i  'j66,  two  memoirs ;  in 
the  first  of  which,  entitled  Elejiienta   Calculi  Variatiouum  (or 

*  "  Method  of  finding  curved  lines  enjoying  the  property  of  maximum  or 
minimum." 

f  The  preceding  account  has  been  taken  almost  entirely  from  Woodhouse's 
Isoperimetrical  Problems;  but  for  what  follows  we  are  indebted  chiefly  to 
Todhunter's  History  of  the  Progress  of  the  Calculus  of  Variations  during  the 
Nineteenth  Century. 


54^  CALCULUS   OF  VARLATLONS. 

Elements  of  the  Calculus  of  Variations),  he  first  gives  our 
science  its  present  name  ;  while  in  the  second  he  enunciates 
the  theorem  of  Prob.  LXVIII. :  this  being  apparently  the 
first  investigation  ever  made  relative  to  the  conditions  of  in- 
tegrability.  Subsequently  in  his  Integral  Calculus,  1770,  he 
extended  the  theorem  to  two  dependent  variables,  as  in  Prob. 
LXX.;  while  Lexel,  in  1771,  established  the  principle  of  Prob. 
LXIX. 

473.  Prior  to  this  period  Lagrange,  w^ho  is  commonly  re- 
garded as  the  author  of  the  calculus  of  variations,  had  com- 
menced his  labors.  But  as  we  have  not  space  to  consider  his 
writings  in  detail,  we  shall  merely  indicate  the  particulars  in 
which  he  improved  our  science. 

First.  Much  ambiguity  and  awkwardness  had  previously 
arisen  from  the  want  of  a  good  method  of  distinguishing  be- 
tween ordinary  differentials  and  those  differentials  or  incre- 
ments which  we  now  call  variations.  This  difficulty  Lagrange 
overcame  by  the  invention  of  the  symbol  d,  which,  like  dy 
could  denote  either  an  increment  or  an  operation,  and  proved 
of  the  highest  importance. 

Secondly.  The  formula  M  =z  o,  and  others,  had  been  de- 
rived by  Euler  from  geometrical  conceptions  by  breaking  the 
integral,  or  the  required  curve,  into  parts,  and  operating  labo- 
riously upon  two,  three,  or  four  consecutive  elements.  But 
Lagrange,  by  deriving  these  formulae  by  the  methods  now  in 
use,  shortened  the  processes  of  obtaining  them,  and  placed  our 
science  upon  its  true  analytical  basis. 

Thirdly.  The  formulae  of  Euler  determined  merely  the 
nature  of  the  required  curve,  its  extremities  being  supposed 
to  be  fixed.  But  Lagrange,  in  what  he  termed  the  definite 
equations,  first  gave  the  form  and  the  interpretation  of  all 
those  formulae  which  are  still  employed  when  the  extremities 
of  the  required  curve  are  not  fixed,  and  which  we  have  called 
the  equations  or  terms  at  the  limits. 


HISTORICAL   SKETCH.  S47 

Fourthly.  Lagrange  invented  that  general  method  which 
is  still  employed,  and  known  as  the  Method  of  Lagrange,  and 
which  enables  us  by  the  use  of  one  or  more  indeterminate 
multipliers  to  discuss  those  cases  in  which  the  variables  are 
connected  by  an  implicit  relation  merely  ;  that  is,  by  a  differ- 
ential equation  which  is  not  integrable. 

Lastly.  Lagrange  first  attempted  to  extend  the  calculus  of 
variations  to  the  case  of  double  integrals.  This  he  did  by 
discussing  Prob.  LVIL,  obtaining  our  equation  (lo),  Art.  366, 
without  considering  the  terms  at  the  limits. 

4-74.  In  the  year  18 10  appeared  an  English  work,  entitled 
"  A  Treatise  on  Isoperimetrical  Problems  and  the  Calculus  of 
Variations,"  by  Robert  Woodhouse,  A.M.,  F.R.S.,  Fellow  of 
Caius  College,  Cambridge.  The  first  five  chapters  of  this 
work,  which  is  a  small  octavo  of  154  pages,  with  9  pages 
of  preface,  are  devoted  to  a  careful  history  of  the  subject  to 
the  time  of  Lagrange,  and  are  all  that  are  now  of  any  in- 
terest, the  remaining  three  containing  little  history  of  im- 
portance. 

The  subject  having  been  next,  but  not  very  successfully, 
treated  by  Lacroix  in  his  Traits  du  Calad  Diffei-enticl  et  du  Cal- 
cul  Integral,  second  volume,  second  edition,  18 14,  there  ap- 
peared two  German  works.  The  former,  by  E.  H.  Dirksen, 
which  is  a  small  quarto  of  243  pages,  with  8  pages  of  preface, 
was  published  at  Berlin  in  1823,  and  is  entitled  "Analytical 
Exhibition  of  the  Calculus  of  Variations,  with  the  Applica- 
tion of  it  to  the  Determination  of  Maxima  and  Minima." 
The  latter  is  entitled  "  The  Theory  of  Maxima  and  Minima," 
by  Dr.  Martin  Ohm,  Berlin,  1825,  and  is  an  octavo  of  330,  with 
a  preface  of  18  pages. 

None  of  these  works,  however,  extended  the  calculus  of 
variations,  and  we  now  resume  the  history  of  its  progress. 

475.  The  first  discussion  of  the  discrimination  of  maxima 
and  minima  appears  to  have  been  undertaken  by  Legend  re, 


548  CALCULUS  OF  VARIATIONS. 

who,  about  the  year  1787,  elaborated  the  method  already  men- 
tioned in  Art.  187,  and  which  was  published  the  following 
year  in  the  History  of  the  Royal  Academy  of  Science.  This 
method  was  subsequently  adopted  by  Lagrange,  although  he 
indicated  the  defect  noticed  in  the  above  article.  This  method 
is  explained  in  Todhunter's  History  of  the  Calculus  of  Varia- 
tions. 

Legendre  seems  also  at  the  same  time  to  have  given  the 
first  instance  of  a  discontinuous  solution  by  showing  in  the 
discussion  of  Prob.  XV.  that  it  might  be  necessary  for  the  re- 
quired curve  to  be  in  part  rectilinear. 

In  the  Memorie  delV  Istituto  Nazionale  Italiano^  Vol.  II. 
Part  II.,  Bologna,  18 10,  Brunacci  extended  the  method  of 
discrimination  to  the  case  of  a  double  integral ;  and  although 
his  method  is  open  to  the  same  objection  as  that  of  Legendre 
for  single  integrals,  he  succeeded  in  establishing  all  the  con- 
ditions relative  to  Fpp,  f^g  and  F^g  mentioned  in  Art.  431; 
and  their  discovery  appears  to  be  due  to  him. 

476.  The  variation  of  a  double  integral  when  the  limits  are 
also  variable,  the  exhibition  of  the  terms  at  the  limits  so  as  to 
determine  the  conditions  which  must  there  hold,  and  the  vari- 
ation of  a  multiple  integral  in  general,  were  subjects  which 
had  not  yet  been  investigated,  and  they  next  engaged  the  at- 
tention of  mathematicians. 

Three  memoirs  were  pubhshed  bearing  more  or  less 
directly  upon  these  subjects :  the  first  by  C.  F.Gauss  in  1829, 
the  second  by  Poisson  in  1 831,  and  the  third  by  Ostrogradsky 
in  1834.  But  while  these  writers  effected  much,  they  did  not 
succeed  in  determining  in  a  general  manner  the  number  and 
form  of  the  equations  which  must  subsist  at  the  limits  in  the 
case  of  a  double  or  triple  integral. 

477.  In  the  seventeenth  volume  of  Crelle's  Mathematical 
Journal,  1837,  appeared  a  memoir,  entitled  "On  the  Theory 
of  the  Calculus  of  Variations  and  of  Differential  Equations,"  by 


HISTORICAL   SKETCH,  549 

C.  G.  Jacobi.  This  memoir,  which  purports  to  be  an  extract 
from  a  letter  to  Professor  Enke,  is  devoted  partly  to  the  cal- 
culus of  variations  and  partly  to  dynamics.  In  the  first  part 
Jacobi  elaborated,  but  without  demonstration,  the  theorem 
which  bears  his  name ;  that  is,  he  assumed  the  truth  of  our 
lemmas,  not  even  giving  the  forms  of  the  functions  A,  A^,  A^, 
B^,  etc.,  although  he  determined  the  form  of  u,  and  merely 
touched  upon  the  connection  between  u  and  v.  See  Art.  174. 
This  brevity  rendered  the  theorem  the  subject  of  numerous 
commentaries,  as  we  shall  presently  see. 

Jacobi  also  touched  upon  the  mode  of  transforming  the 
terms  of  the  first  order  in  the  variation  of  a  double  integral, 
but  effected  nothing  of  importance. 

4-78,  In  1 841  were  published  three  memoirs  relative  to 
Jacobi's  theorem  :  the  first  two  by  V.  A.  Lebesgue  and  C. 
Delaunay,  in  the  sixth  volume  of  LiouwiWe's  /oicrna/  0/  Mat ke- 
mattes,  and  the  last  by  Bertrand  in  Xh^  Journal  de  V Ecole  Poly- 
techniqite.  The  proof  given  by  Delaunay  is  that  which  we 
have  followed  in  our  notes  to  Lemmas  I.  and  II.,  and  he  has 
been  generally  followed  by  subsequent  writers. 

479.  As,  notwithstanding  the  labors  of  Gauss,  Poisson, 
Ostrogradsky  and  Jacobi,  no  general  method  of  treating  the 
terms  at  the  limits  in  the  case  of  multiple  integrals  had  yet 
been  discovered,  the  Academy  of  Science,  Paris,  1842,  pro- 
posed for  its  mathematical  prize  the  following  subject:  To 
find  the  limiting  equations  which  must  be  combined  with  the 
indefinite  equations  in  order  to  determine  completely  the 
maxima  and  minima  of  multiple  integrals ;  the  formulas  to  be 
applied  to  triple  integrals. 

Of  the  four  memoirs  presented,  that  by  Sarrus  was  ad- 
judged worthy  of  the  prize,  while  that  by  Delaunay  received 
honorable  mention ;  the  examiners  being  Liouville,  Sturm, 
Poinsot,  Duhamel  and  Cauchy. 

The  memoir  of  Sarrus  is  entitled  Recherches  sur  le  Calcul 


550  CALCULUS  OF  VARLATLONS. 

des  Variations^  and  may  be  found  in  the  tenth  volume  of  the 
Savants  Etr angers,  1846,  and  occupies  127  quarto  pages.  By 
means  of  his  new  symbol  of  substitution,  Sarrus  may  be  said 
to  have  solved  the  problem  proposed  by  the  Academy,  and 
his  memoir  is  one  of  the  most  important  contributions  of  the 
century.  But  this  sign  of  substitution  as  invented  by  Sarrus, 
besides  having  an  inconvenient  form,  signified  merely  the  sub- 
stitution of  a  particular  value  of  a  variable  for  its  general 
value,  and  his  method  therefore  lacked  brevity. 

The  treatment  by  Delaunay  is  much  less  general,  assuming 
that  in  the  case  of  double  integrals  the  limiting  cylinder  or 
surface  is  to  be  continuous  and  closed.  His  memoir  was  pub- 
lished in  the  29th  cahier  of  the  Journal  de  V Ecole  Polytech- 
niqiie,  dated  1843,  ^^^d  seems  to  have  been  followed  by  all  the 
writers  on  the  calculus  of  variations  subsequent  to  Moigno 
and  Lindelof. 

4-80.  The  next  advance  was  made  by  Cauchy  in  a  memoir 
on  the  calculus  of  variations  published  in  the  third  volume 
of  his  Excrcices  d' analyse  et  de  Physique  Mathhnatiqiie,  1844, 
extending^  from  page  50  to  page  130.  This  memoir  is  little 
else  than  a  reproduction  of  the  investigations  of  Sarrus,  but 
in  it  Cauchy  effected  much  of  the  needed  condensation  by 
giving  to  the  sign  of  substitution,  like  that  of  integration  in  a 
definite  integral,  the  power  of  denoting  substraction  also, 
while  its  form  was  changed  to  that  which  we  have  adopted. 
For  further  particulars  regarding  this  part  of  our  subject  the 
reader  who  does  not  wish  to  examine  the  original  memoirs 
may  consult  the  chapters  on  Sarrus  and  Cauchy  in  Todhun- 
ter's  History  of  the  Calculus  of  Variations. 

48(,  We  must  next  notice  some  systematic  treatises  which 
now  appeared. 

As  a  successor  of  Woodhouse  there  appeared  "  A  Treatise 
on  the  Calculus  of  Variations,"  by  Richard  Abbatt,  London, 
1837.     This  is  an  octavo  of  207  pages,  with  11  pages  of  pre- 


HISTORICAL   SKETCH.  551 

face,  but  is  of  no  great  importance  at  the  present  day,  and 
could  hardly  be  regarded  as  a  complete  treatise. 

In  the  year  1850  appeared  a  work  entitled  ''  An  Elementary 
Treatise  ori  the  Calculus  of  Variations,"  by  the  Rev.  John 
,  Hewitt  Jellett,  A.M.,  Fellow  of  Trinity  College  and  Profes- 
sor of  Natural  Philosophy  in  the  University  of  Dublm.  This 
work,  which  is  an  octavo  of  377  pages,  with  an  introduction 
and  preface  of  20  pages,  is  one  of  the  most  important  which 
have  appeared  in  any  language,  and  is  not  elementary  as  its 
title  would  imply.  But  Prof.  Jellett  had  not,  as  he  himself 
tells  us,  been  able  to  peruse  the  memoir  of  M.  Sarrus,  while 
that  of  Cauchy  is  not  mentioned  by  him  at  all.  Hence  his  dis- 
cussion of  multiple  integrals,  in  which  he  follows  that  memoir 
of  Delaunay  which  received  honorable  mention  by  the  French 
Academy,  is  defective,  and  cannot  be  recommended  to  the 
student. 

Ohm's  treatise  was  succeeded  by  a  voluminous  work  by 
Dr.  G.  W.  Strauch,  entitled  Theorie  und  Anwendung  des  soge- 
nannten  VariationscalcuV s^  Zurich,  1849.  This  treatise  consists 
of  two  closel}^  printed  large  octavo  volumes,  the  first  contain- 
ing 499  pages,  with  32  pages  of  preface,  and  the  second  788 
pages  ;  and  is  chiefly  valuable  for  its  great  number  of  carefully 
solved  examples,  and  historical  notes,  although,  as  might  be 
expected,  much  of  the  matter  has  little  or  no  connection  with 
the  calculus  of  variations.  Strauch  does  not  exhibit  the 
theorem  of  Jacobi,  although  he  generally  examines  the  terms 
of  the  second  order,  employing  the  method  of  Legendre  and 
Lagrange  without  even  noticing  its  defect.  He  is  also  like 
Jellett  deficient  in  the  treatment  of  multiple  integrals,  not  fol- 
lowing the  method  of  Sarrus  and  Cauchy.  Strauch  sub- 
sequently, in  1856,  presented  to  the  Academy  of  Sciences  in 
Vienna  a  memoir  entitled  Anwendujig  des  sogenannten  Varia- 
tionscalcuV s  auf  zzvcifache  und  dreifache  Integrale,  and  pub- 
lished in  the  i6th  volume  of  the  Denkschriftc7i  of  the  Acad- 
emy, 1859,  where  it  occupies  156  large  quarto  pages;  and  in 


552  -    CALCULUS  OF  VARLATLONS. 

this  memoir  he  even  declares  that  Sarrus  and  Cauchy  did  not 
solve  the  problem,  proposed  by  the  French  Academy.  His 
own  memoir  is,  however,  of  no  importance. 

In  a  few  years  appeared  another  German  w^rk  by  Dr. 
Stegmann,  entitled  Lehrbuch  der  Variationsrechnung  und  ihrer . 
Anwendimg  bei  Untersuchungen  fiber  das  Maxiinum  und  Mini- 
mum^ Kassel,  1854.  This  is  an  octavo  of  417  pages,  with  16 
pages  of  preface,  but  is  not  so  rich  in  examples  as  is  the 
treatise  by  Strauch,  while  it  possesses  the  two  defects  men- 
tioned in  connection  with  that  treatise. 

Prof.  Bruun  published  in  the  Russian  language  "  A  Manual 
of  the  Calculus  of  Variations,"  Odessa,  1848,  which  is,  accord- 
ing to  Prof.  Todhunter,  an  octavo  of  195  pages. 

We  may  mention,  finally,  that  Prof.  Price  in  the  second 
volume  of  his  Treatise  on  Infinitesimal  Calculus,  Oxford, 
1854,  devoted  more  than  100  pages  to  our  science,  explaining 
the  theorem  of  Jacobi,  and  touching  upon  the  subject  of 
double  integrals. 

482.  After  the  publication  of  the  three  memoirs  mentioned 
in  Art.  478,  the  subject  of  the  discrimination  of  maxima 
and  minima  was  not  considered  for  about  ten  years,  after 
which  it  was  resumed  earnestly  by  mathematicians  in  papers, 
some  of  which  we  will  next  mention. 

In  the  third  volume  of  Tortolini's  Annali  di  Scienze  Mathe- 
matiche  e  Fisiche,  1852,  appeared  an  article  of  more  than  40 
pages  by  Prof.  G.  Mainardi,  claiming,  but  without  good  rea- 
son, to  exhibit  a  new  method  of  discriminating  maxima  and 
minima.  But  he  also  extended  Jacobi's  theorem  to  double 
integrals,  and  his  method  has  been  followed  by  us  in  treating 
this  subject. 

In  the  same  volume  appeared  a  short  article  on  the  same 
subject  by  Prof.  F.  Brioschi.  Mainardi  had  indicated  the 
value  of  the  theory  of  determinants  in  connection  with  the 
exhibition  of   the   terms  of   the  second  order,  and  Brioschi 


HISTORICAL    SKETCH.  553 

employed  it  freely,  this  being  apparently  the  first  attempt  to 
apply  determinants  to  this  subject. 

There  next  appeared  a  quarto  pamphlet  of  20  pages 
regarding  Jacobi's  theorem,  entitled  Untersiichungen  ilber  Varia- 
tions-re chnung.  Inaugural-Dissertation  von  Dr.  Friedrich  Eisen- 
lohr,  Manheim,  1853. 

The  subject  was  next  considered  in  a  work  entitled  "  On 
the  Criteria  for  Maxima  and  Minima  in  Problems  of  the  Cal- 
culus of  Variations,"  which  was  presented  by  Spitzer  to  the 
Academy  of  Sciences  at  Vienna  in  1854.  This  work  consists 
of  two  memoirs  occupying  together  more  than  135  pages,  the 
first  being  published  in  the  12th  and  the  second  in  the  14th 
volume  of  the  Sitziingsberichte  of  the  Academy,  and  to  these 
memoirs  we  are  indebted  for  the  exceptions  which  we  have 
noticed  in  connection  with  Jacobi's  theorem.  But  Mainardi 
and  Spitzer  did  not  confine  themselves  to  the  development  of 
Jacobi's  theorem,  but  sought  rather  to  establish  new  methods 
of  their  own,  both  of  which  are,  according  to  Prof.  Todhunter, 
"  Legendre's  method  improved  by  additions  borrowed  from 
Jacobi." 

In  the  54th  volume  of  Crelle's  Mathematical  Journal^  1857, 
appeared  a  memoir  by  Otto  Hesse,  entitled  "  On  the  Criteria 
for  the  Maxima  and  Minima  of  Single  Integrals,"  extending 
over  pages  227-273.  Hesse  confines  himself  exclusively  to 
the  application  of  Jacobi's  theorem  to  smgle  integrals  involv- 
ing only  one  dependent  vajriable,  and  his  memoir  is  the  most 
elaborate  which  has  yet  appeared  regarding  this  subject. 
See  Arts.  184,  186. 

In  the  55th  volume  of  Crelle's  Mathematical  Journal,  1858, 
appeared  a  memoir  by  A.  Clebsch,  entitled  ''  On  the  Reduction 
of  the  Second  Variation  to  its  Simplest  Form,"  and  extending 
over  pages  254-273.  The  object  of  Clebsch  was  to  general- 
ize the  theorem  of  Jacobi,  and  to  supply  investigations  like 
those  of  Hesse  for  the  case  in  which  the  single  integral  con- 
tains several  dependent  variables  with  or  without  connecting 


554  CALCULUS   OF  VARLATLONS. 

equations,  and  also  for  multiple  integrals.  The  former  point 
had  not,  so  far  as  the  author  knows,  been  hitherto  discussed, 
but  the  latter  had  been  considered  by  Mainardi.  The  subject 
of  multiple  integrals  is  resumed  by  him  in  a  third  memoir, 
entitled  '*  On  the  Second  Variation  of  Multiple  Integrals,"  and 
published  in  the  56th  volume  of  Crelle's  Mathematical  Jojir- 
naly  1859,  where  it  extends  over  pages  122-148.  His  second 
memoir  is  "  On  those  Problems  in  the  Calculus  of  Variations 
which  involve  only  one  Independent  Variable,"  and  is  in  the 
same  volume  which  contains  his  first  memoir. 

483.  We  now  come  to  a  most  valuable  work,  enti- 
tled ''A  History  of  the  Progress  of  the  Calculus  of  Varia- 
tions during  the  Nineteenth  Century,"  by  I.  Todhunter, 
M.A.,  Fellow  and  Principal  Mathematical  Lecturer  of  St. 
John's  College,  Cambridge.  Macmillan  &  Co.,  London,  1861. 
This  volume  is  a  large  octavo  of  530  pages,  with  10  pages  of 
preface,  and,  taken  together  with  the  first  five  chapters  of 
Woodhouse,  furnishes  a  complete  history  of  our  subject. 
But  in  addition  to  the  mathematics  necessary  to  the  histori- 
cal sketches,  much  of  which  has  been  superseded  by  bet- 
ter methods,  Prof.  Todhunter  has  frequently  introduced  these 
better  methods,  and  has  given  such  other  investigations  of  his 
own  that  his  work  contains  nearly  all  the  matter  necessary  to 
form  a  modern  treatise,  although,  from  the  nature  of  the  case, 
it  is  so  arranged  as  to  be  of  little  service  to  the  reader  who  is 
not  already  tolerably  familiar  with  the  calculus  of  variations. 
We  append  the  subjects  of  the  seventeen  chapters :  Chap.  I., 
Lagrange,  Lacroix  ;  II.,  Dirksen,  Ohm  ;  III.,  Gauss  ;  IV.,  Pois- 
son  ;  v.,  Ostrogradsky  ;  VI.,  Delaunay  ;  VII.,  Sarrus ;  VIII., 
Cauchy  ;  IX.,  Legendre,  Brunacci,  Jacobi ;  X.,  Commentators 
on  Jacobi  ;  XL,  On  Jacobi's  Memoir  ;  XIL,  Miscellaneous 
Memoirs ;  XIIL,  Systematic  Treatises ;  XIV.,  Minor  Treatises ; 
XV.',  XVI.,  Miscellaneous  Articles;  XVII.,  Conditions  of 
Integrability.     The  last  chapter  is  a  complete  history  of  the 


HISTORICAL   SKETCH,  555 

subject  from  the  earliest  times  as  it  had  not  been  mentioned 
by  Woodhouse,  nor  had  its  history  been  given  by  any  pre- 
vious writer. 

4-84.  A  few  months  subsequently,  but  during  the  same 
year,  1861,  appeared  the  last  systematic  treatise,  the  Calcul 
des  Variations,  by  Moigno  and  Lindelof.  But  the  title-page  of 
this  work,  which  is  a  small  octavo  of  352  pages,  with  20  addi- 
tional pages  of  preface,  introduction,  etc.,  presents  it  in  the 
beginning  as  merely  the  fourth  volume  of  the  Lemons  de  Calcid 
Differentiel  et  de  Calcul  Integral,  by  M.  I'Abbe  Moigno,  the 
distinctive  title  following  subsequently.  According  to  Moigno, 
the  chief  credit  of  this  work,  which  is  the  only  complete 
treatise  in  the  French  language,  belongs  to  his  colleague,  M. 
Lindelof,  then  a.  young  professor  from  the  university  of 
Helsingfors  in  Finland,  who  had  made  the  calculus  of  varia- 
tions a  specialty,  and  who  gave  Moigno  freely  the  benefit  of 
his  knowledge. 

This  treatise  was  the  first  to  present  a  satisfactory  account 
of  the  conditions  which  must  hold  at  the  limits  when  we  wish 
to  maximize  or  minimize  the  double  or  triple  integral.  But 
although  the  methods  followed  are  substantially  those  of 
Sarrus  and  Cauchy,  the  authors  have,  in  many  cases,  greatly 
simplified  the  formulae  of  their  masters  ;  and  to  this  portion  of 
the  Calcid  des  Variations  the  present  author  is  almost  entirely 
indebted  for  the  discussions  which  have  been  presented  in 
Chapter  III.  ;  although  the  view  of  variations  adopted  by 
Moigno  and  Lindelof  is  that  followed  by  Sarrus,  and  which 
has  been  explained  in  Section  VL  Chap.  IIL 

485.  It  had  long  been  known  that  a  discontinuous  solu- 
tion might  become  necessary  in  certain  problems.  But 
although  particular  cases  had  been  discussed  by  Legendre 
and  others,  nothing  resembling  a  general  theory  of  such 
solutions  had  yet  been  propounded. 

In  the  Philosophical  Magazine  for  June,  1866,  Prof.  I.  Tod- 


55^  CALCULUS  OF  VARLATIONS. 

hunter  first  announced  the  principle  that  variations  might 
be  of  restricted  sign,  thus  rendering  it  unnecessary  for  the 
equation  M  —  o  \.o  hold  throughout  C/;  and  this  may  be 
regarded  as  the  fundamental  principle  of  the  theory  in  ques- 
tion. This  discovery  appears  to  have  been  due  mainly  to  the 
difficulties  presented  by  the  consideration  of  Prob.  XVI. 

In  1869  this  subject  was  proposed  at  Cambridge  for  the 
Adams  Essay,  and  elicited  from  Prof.  Todhunter  in  1871  the 
prize  essay,  which,  with  slight  alteratfon,  was  published  in  the 
same  year  by  Macmillan  &  Co.,  under  the  title  "  Researches 
in  the  Calculus  of  Variations." 

This  work,  which  is  an  octavo  of  278  pages,  and  8  pages 
of  introduction,  is  certainly  the  most  important  original  con- 
tribution which  our  science  has  received  since  the  appearance 
of  the  essay  of  Sarrus,  inasmuch  as,  -in  it,  the  author,  while 
discussing  incidentally  many  other  points  of  interest,  did  for 
the  theory  of  discontinuous  solutions,  what  Sarrus  did  for  that 
of  multiple  integrals.  The  case  of  single  integrals  only  is  dis- 
cussed, and  these  are,  with  a  few  exceptions,  supposed  to 
involve  but  one  dependent  variable.  The  theory  is,  how- 
ever, abundantly  illustrated  by  examples ;  and  we  cannot  too 
strongly  recommend  the  work  to  our  readers,  since,  from  it, 
we  have  derived  most  of  what  we  have  presented  in  Section 
IX,  Chap.  I. 


NOTES. 


NOTE  TO   LEMMA   I. 

To  establish  this  theorem,  which  belongs  entirely  to  the  differential  calculus, 
we  shall  employ  the  symbolic  language,  or,  as  it  is  sometimes  called,  the  calculus 
of  operations.  (See  De  Morgan's  Diff.  and  Integ.  Calc,  page  751;  also  Boole's 
Diff.  Eqs.,  Chap.  XVI.) 

Let  d  denote  differentiations  with  respect  to  k  only,  and  D  with  respect  to  8y 
only,  both  k  and  8y  being  regarded  as  functions  of  x,  and  the  differentials  with  re- 
gard to  X  being  total.  Then  any  order  of  total  differential  of  any  function  of  k  dy 
may  be  written  {d-\-  DY  of  that  function.     Now  putting  v  for  pair  (4),  we  have 

v=\{d-\-  DY  D^  ±  {d-\-  DY  D^lk  8y 

=  id -{•  ny  D\{d -\- D)"'-^  ±  D^'-^kdy 

=  X\{d-^D)^-^  ±  D^-^k  Sy,  (I) 

where 

X={d-ir  D)D  =  dD  +  D\  (2) 

It  must,  however,  be  remembered  that  ^does  not  denote  a  quantity,  but  merely  a 
mode  of  differentiation,  and  that  seeming  exponents  as  2,  n,  etc.,  do  not  indicate 
powers,  but  the  number  of  times  that  a  certain  mode  of  differentiation  is  per- 
formed. 

Now  from  (2)  we  have 

Z?2  4-^Z)  +  ^  =  ^_|_X=-(^'^  +  4^),  (3) 

4         4  4 

or 


55^  CALCULUS  OF  VARIATIONS. 

the  first  member  of  (4)  denoting  differentiation  twice  according  to  a  predetermined 
method,  the  second  differential  having  been  rendered  perfect  by  the  addition  of 
another  differential,  just  as  the  square  is,  in  quadratics,  by  the  addition  of  a  square. 
Hence,  solving  as  in  quadratics,  we  obtain 

Z)  =  I|_^±  (^2  +  4^)1  I  (5) 


and 


d^D  =  l\d±{d'-^^^Xf  \^\{d±r\  (6) 


r  dqpoting  also  a  mode  of  differentiation  only. 

Now  put  n  for  m  —  /.     Then  in  (i),  by  the  use  of  (5)  and  (6),  we  have 

{d-\-  Dr-^  ±  n^-i  =  (^+  ny  ±D^  =  \  \{d±rY  ±{-d±  rf  \  , 


(7) 


the  positive  or  negative  sign  being  used  according  as  n  is  even  or  odd. 

If  we  first  suppose  n  to  be  even,  and  expand  both  binomials  by  the  binomial 
theorem,    and    add   the    results,    then,    since    each   term  which  does  not  cancel 

becomes  double,  we  shall,  after  multiplying  —  by  2,  have 


dn  _|_  '!kL_l\  dn- 1  ^2  _|_  etc.    ^  .  (8) 


Let  us  next  suppose  n  to  be  odd.  Then  the  development  will  assume  the  same 
form,  because  the  sign  connecting  the  two  binomials  will  be  negative,  so  that 
we  shall  have  always 

(^  +  Z))— ^± /)—'::=  ^J«'~  +  ^^^^^^"- V2-f- etc.  I  .  (9) 

But  since  r^  =  d^ -^  4X,  we  will  suppose  the  values  of  r\  r\  etc.,  in  (9)  to  have 
been  found  and  arranged  according  to  the  ascending  superscripts  of  X.  Then 
there  will  occur  in  r^,  r*,  r^,  etc.,  one  term  involving  d,  but  not  X;  and  this  term, 
when  combined  with  its  component  outside  of  r,  which  will  involve  d,  will  always 
become  d»  multiplied  by  some  function  of  n.  Therefore  the  development  may  be 
written  4^ 

(d-^D)'^-^  ±  B'^-i  =  ad^ -{-  dd^-'^ X -{-  cd^-"^ X^  -}-  etc.,  (10) 

where  a,  d,  c,  etc.,  are  functions  of  n  and  numbers  merely.  Hence  from  (i)  v^e 
have 


NOTES.  559 

We  have  now  only  to  abandon  the  symbols  of  separation  by  performing  the 
operations  which  they  indicate,  thus:  , 


■^       dx^         dx^  ^   dx^ 

and  since  d-\-  D  denotes  total  differentiation,  we  have 


^^5/0; 


(12) 


{d^D)^ci8yfJ)  =  ^^ci8yif). 


Proceeding  in  like  manner  with  the  other  terms  in  (ii),  we  shall  finally  obtain 


d^  ^?  +  i 


where 


2,nd 


ci  =  — —  ak, 
dx'^ 


O+i 


dx'^- 


hh 


<:i+i  = 


dx^-^ 


ck, 


(13) 


(14) 


-      ^      i  r  _L  ^(^  -  ^)    I    ^(^  -  ^){n  -  2){n  -  3)   ,  )      1 

^(^?  -  I)  _L  8  ^^(^  -  i)(^?  -  2){n  -  3)       ^^^    ) 


b  = 


c  =  -^-  i  i6 

2n  — 1     ' 


^(n  —  j)(n  —  2)(n  —  3) 


+  etc. 


(15) 


Now  the  application  of  (13)  is  in  reality  simple.  For  we  see  that  for  any  given 
value  of  n  its  number  of  terms  must  always  be  the  same  as  that  of  equation  (9). 
Hence  there  will  be  but  one  term  when  «  is  i,  two  when  w  is  2  or  3,  three  when 
«  is  4  or  5,  etc.  Moreover,  it  will  be  found  from  (15)  that  a  is  always  unity,  and 
that  5  =  n  when  n  has  any  value  from  2  to  5  inclusive,  and  that  c  =  2  when  n  is 
4,  and  is  5  when  «  is  5. 


^     560  CALCULUS   OF  VARIATIONS. 


NOTE  TO   LEMMA   II. 


The  integration  may  be  effected  as  follows:  Multiply  equation  (i)  by  ut,  and 
subtract  the  product  from  (2).     Then  we  have 

« 

d  d^  d  d'^ 

U  =u—  Ai{ut)' -\-  u  —J  A'i{ut)"  4-etc.  —  ut—  Aiu'  —  ut-—Aiu"  —  etc. 

* 

(       d™-  d™-  1 

Now  we  know  that  if  F  and  Q  be  any  two  quantities,  we  shall  have 

pQin)  :=  (pQyn)  _  ;^(p'0(«-l)  _|_  ''^''   ~  1^  (/'"0(«-2)  _  etC.  (2) 

For  let  d  denote  differentiations  with  respect  to  Q,  and  D  with  respect  to  F.    Then 
we  shall  have 

PQ(.n)  ^  d^FQ,         {FQf^)  =  {d-{-  DYFQ,         (P'0(«-i)  =  {d -\.  Z?)«-i  DFQ,  etc. 

Now  in  the  cases  which  we  shall  consider  n  will  be  a  not  large  positive  integer, 
and  it  will  therefore  readily  appear  by  trial  that 

d^  =  (d-\- BY  -  n{d-\- I)Y-^D-\-''^'^~^^ (^+  Z))«-2/)2  _  etc.  ±  i>. 

Hence  if  we  select  any  term  of  (i),  as 

d™' 
u-^Am{uti'^'>    or    ulAm{utY'^')Y^'^, 

and  put  u  for  F,  and  the  other  factor  for  ^"^we  obtain,  by  the  use  of  (2), 
u  [An,  («/)W]W  =  [uAr^  (w^)^]^  _  m  [u'Am  (z//)('")]('"-i) 

+  ^(^  -  ^iru"Ar.  (uf)('-)](^-^)  -  etc.  (3) 

But 


(«/)(»»)  =  «/(»«)+  W«7(«-l)  +  ^ ^  «"/(m-2)_|_ 


etc.  (4) 


NOTES.  561 

Substituting  this  value  in  (3),  each  term  may  expand  into  a  series,  each  series 
having  the  superscript  of  the  term  from  which  it  was  derived.  Now  consider 
any  series,  and  let  its  superscript  be/,  so  that  it  must  be  the  m  —  p -\- \  xn  order, 
and  every  term  must  contain  the  factor  u^"^~p^  Am.  Now  for  an  individual  term, 
take  that  in  this  series  whose  order  is  m  —  q-\-i,  or  q  -\-  i  when  we  begin  at  the 
last.     This  term  will  be  of  the  form 


where  k  =  u^-'^-p^  Am  «("*-?>,  and 


(5) 


^^^  _^  ^^(^-  I)-  •  •  ■  (/+!)      mjm-i) (^4-1) 

I,  2,  3  ....  (w  —J>)  I,  2,  3  .  .  .  .  (?n  —  q)  ' 

while  /  and  q  must  be  some  positive  integer,  or  zero,  and  m  some  positive  integer. 
Now  Up  and  q  be  unequal,  there  will,  supposing/  greater  than  q,  certainly  arise 
in  the  series  whose  superscript  is  ^  a  term  of  the  form 


dod  \    dxP  i 

the  signs  being  like  or  unlike  according  asp  —  q  is  even  or  odd.  '  Hence,  by  the 

theorem  of  the  preceding  note,  all  the  terms  in  2u  -—  Am  {uff^"'  in  which/  and  q 

are  unequal  may  be  transformed,  so  that  by  adding  those  in  which  p  and  q  are 
equal,  which  have  already  the  required  form,  we  may  write 

^«  ^  ^-  («^>'"^  =  ^'+  ^  ■^^''  +  ^  ^^'"  +  "'"•  ^7) 

But  if  the  dJ£ferentiation  indicated  in  the  first  member  of  (7)  were  performed,  it  is 
evident  that  the  terms  which  would  contain  /  undifferentiated  would  be 

^u  {Am  «<'"))('»)  t=Bt=z  ^ut  {Am  «<'"))('»).  (8) 

Hence  it  appears  from  (i)  that  all  the  terms  containing  t  undifferentiated  will 
disappear  from  U,  and  we  shall,  therefore,  have 


^=l^-''+;^^^'"+"=-  fe' 


Therefore 


/ 


d 
Udx  =  Bit'  -\ B-i  t"  A-  etc.  (10) 

dx 


5^2  CALCULUS  OF  VARIATIONS. 


NOTE   TO   ART.  369. 

Let  A,  B  and  C  denote  the  angles  made  by  the  normal  with  x,  y  and  z  respec- 
tively, and  let  the  Greek  letter  |  (xi  or  x)  denote  the  angle  made  with  the  plane 
of  xz  by  the  plane  which  contains  the  normal  and  is  parallel  to  z,  and  r}  (eta  or  e) 

c 

be  /  tan  —  or  I  tan  c,  so  that,  e  being  the  Napierian  base,  we  shall  have 

C 

en  =  tan  —  =  tan  c. 

2 

Our  object  is  now  to  change  in  (10)  the  independent  variables  from  x  and  >/  to  ^ 
and  77.     We  have 

.     ^        •  .  2  sin  r       „  2  tan  c  2ev 

sm  C  =  sm  2c  =  2  sm  ^  cos  <r  = cos^^r  =  — ; —  — — ---,  (i) 

cos^  I  +tan'-^r       1  +  ^27,  '  ^V 


(cosV         \ 
-:— i I     sin^^ 
smv         j 


tanV  _  e—'m  —  i 

I         "  ^-2i?4-  I 

'  tanV 


(2) 


These  equations  will  enable  us  to  express  sin  C  and  cos  C  in  terms  of  cos  rf  and 
tan  77;  but  the  process  will  evidently  involve  the  theory  of  hyperbolic  sines, 
cosines,  etc.     We  may  recall  from  this  theory  the  following  formulae,  putting  i  for 

I                                              I                                                 ^^*"  —  I 
sin  u  =  —  (<?»■«  —  <?~*"),         cos  u—  -  (,?'«  -f-  e-»"),         i  tan  u  =    ^^^ (3) 

These  formulae  occur  in  De  Morgan's  Diff.  and  Integ.  Calc,  pages  114  and  119, 
except  that  we  have  put  u  for  0  on  the  first  page,  and  for  x  on  the  second.  Now 
if  in  the  second  and  third  of  these  equations  we  put  irf  for  «,  we  shall  have 

I  /           .      N      ^^'J+i  •  •         e-'»i—i  .  . 

costn  =  -  U-v4-eri)  = ■ — ,        z  tan  ir/  =  — - — j —  (4) 

Hence  from  (i)  and  (2)  we  have 

sin  C  = r-,         cos  C  =  i  tan  in.  (5) 

cos  trj 


NOTES.  563 

Now  it  is  evident  that 

cos  ^  =  sin  C  cos  6  = -^^ ,         cos -5  =  sin  C  sin  |  = (6) 

cos  trf  cos  IV 

Hence  equation  (10)  Art.  366,  now  becomes 


d  cos  h  _,d_  sin  %   __ 
dx  cos  i7]       dy  cos  ir} 


-sin4i  +  cos^|-  +  aan.-^|cos^g  +  sin^J[=o, 


(7) 


in  which  we  must  next  determine  the  values  of  the  partial  differential  coefficients 

d^    dc,    drj        ,   dri 
-J-,  --,  --  and  --• 
dx   dy    ax  dy 

Let  X,  V  and  Z  be  the  co-ordinates  of  any  point  of  any  tangent  plane  to  the 
required  surface,  and  D  the  distance  of  this  pla^ne  from  the  origin.  Then  we 
shall  have 

Jf  cos  ^  +  F  cos  ^  +  Z  cos  C  =r  Z> ; 

and  substituting  in  this  the  values  of  cos  A,  cos  B  and  cos  C,  and  multiplying  by 
cos  ir/,  we  have  t 

Xcos  ^-\-V  sin  q-\-  Zi  sin  ir/  =  D  cos  irj  =  —  C,  (8) 

the  Greek  letter  zeta  being  used  for  convenience  only.  But  since  (8)  represents 
the  equation  of  any  tangent  plane,  and  every  point  of  the  required  surface  lies  on 
one  of  these  planes,  the  equation  of  this  surface  may  be  written 

^         X  cos  k  -\-y  sin  ^  +  zi  sin  t7j  =  —  C,  (q) 

where  C  is  no  longer  constant,  but  must  be  such  a  function  of  ?  and  77  that  the 
variable  Z>  may  always  have  the  meaning  just  assigned.  Moreover,  we  may 
regard  every  point  of  the  required  surface  as  lying  at  the  intersection  of  three 
tangent  planes  drawn  indefinitely  near,  so  that  in  the  second  ^  may  become 
?  4"  ^^.  V  remaining  unchanged  ;  and  in  the  third  7  may  become  7  +  d?/,  ^ 
remaining  unchanged  ;  |  and  7/  themselves  belonging  to  the  first.  Hence  we 
have  a  right  to  differentiate  (9)  with  regard  to  c,  and  7;  separately,  treating  x,  y 
and  z  as  constants.     Performing  this  operation,  we  have 

X  sm  ^  —  y  cos  ?  =  — -,         z  cos  tn  =  —-•  (,10) 

dq  '      drj 


5^4  CALCULUS  OF  VARIATIONS, 

Next  assume,  for  brevity, 

di]       d^^  d^drf  '  dr]   ^   drf     ^    ' 

We  have  also,  from  (9)  and  the  last  of  equations  (10), 

—  {x  cos  I  +/  sin  ^)  =  C  +  zi  sin  iv,         zi  sin  irj  =  i  tan  ir/  — -  •    '       (12) 

We  will  now  differentiate  equations  (10)  and  (11),  reducing  by  means  of  the  equations 
whose  first  members  are  the  bracketed  quantities,  and  supposing  the  last  to  have 
been  obtained  first,  so  as  to  employ  it  in  reducing  the  first.     Thus  we  shall  have 

dx  cos  ^-\-dy  sin  ^=\x  sin  ^  —y  cos  ^]  d^  —  i  sin  irf  dz 

dC  dC 

+  [z  cos  tr/]  dV  —  -^d^  —  --dr] 

=  —  d\zi  sin  i7J\  =^  —  i  tan  irf  (v  d^-\-  w  dff), 

d%  d% 

dxsin  ^  —  dyzos  ^  =  —  [^  cos  ^+_ysin  ^]^|  +  —  d^-\--jr^drj 

d'K  dK 

=  Z,dh  +  [zi  sin  irj]  ^^  +  —  ^|  +  -—-  drj  =  udk-\-  vdrf, 

d'^Z,  d^C, 

dz  cos  i?/  =  [zi  sin  iy]  dr]  +  -ir~^  dk  +-7-Y  drj  —  v  dc, -\-  v)  drj. 


(13) 


d^d^  ^  '  di 

Now  if  in  the  first  two  of  these  equations  we  first  make  dy  zero  in  each  and 
divide  by  dx,  and  then  dx  zero  in  each  and  divide  by  dy,  we  shall  obtain  four 

equations,  the  first  two  of  which  will  each  contain  —  and  — ,  and  the  last  two  of 

which  will  each  contain  — -  and  -7^;  and  these  differentials  will  then  become  the 

dy  dy 

partial  differentials  sought.      Then  finding  the  values  of  these  differentials  by 
common  algebraic  methods,  we  shall  have 

d'q  _  wi  tan  irj  sin  ^  —  v  cos  I 
dx  i  tan  irj  {uw  -\-  v^) 

dr}  _  vi  tan  irf  sin  ^  —  «  cos  I 
dx  i  tan  ir]  {uw  -\-  v^) 

d^  _       wi  tan  irf  cos  |  -}"  ^  sin  g 
dy  ~  i  tan  irf  (uw  -f-  v^) 

drf  _      vi  tan  irf  cos  I  +  «  sin  | 
dy  "  i  tan  irf  {uw  -\-  v^) 


NOTES.  565 

If  now  we  substitute  these  values  in  equation  (7),  observing,  if  we  clear  frac- 
tions, not  to  remove  the  imaginary  quantity  i  from  the  denominator,  it  will  easily 
reduce  to 

«4-Z£/  =  0.  (14) 

This  equation  is  not  itself  integrable,  but  we  can  easily  obtain  from  it  a  more 
general  expression,  which  can  be  integrated. 

By  making  d'\  and  dr]  alternately  zero  in  the  last  of  equations  (13),  we  obtain 

dz         .  dz         .  ,    ^ 

V—  —^  cos  IT},         w  ^=—-  cos  trj,  (15) 

dc,  drf 

where  the  differentials  of  z  have  become  partial,  being  taken  with  relation  to  ^ 
and  77  only,  as  separate  independent  variables.  Now  since  (14)  holds  for  every 
point  of  the  required  surface,  its  differential  with  relation  to  |  or  77,  or  both,  must 
be  zero  also.  Let  us  therefore  differentiate  with  respect  to  rj  only.  Then  ob- 
serving that 

>     d    .         .              ^         i^  „  „  ... 

1  -\-  -J- 1  iSiniTf  =  1  A — —  =  I  —  sec2 in  =  —  tan^ in  =  ttamn't  tan  ?w, 


we  have 


^«       dZ,     ,     d    .         .  ,   ,    .         .    dK   ,      dK 
-T-  =  -7-  (i  +  —  ?  tan  t?/)  -f- 1  tan  tt] 


dri      dr}^     '  df}  "  '  '  drf   '   d^'^drj 

=  .tan.^(nan.;;-+— )  +^^^  =-   +z..tanev 


dH  .         dz   ,   , 


~  T^F  ^°^  ^'^'vUZJ  ^^"^  ^^' 


div        d'^z  .  dz 

-J—  =  TT  cos  t77 —t  sin  in. 

drj        drf  '      drf  ' 

Hence,  by  adding,  we  deduce  from  (14) 

d'^z    ,    d'^z  ,  r. 

dz^       drj^ 

a  partial  differential  of  the  second  order,  the  complete  integral  of  which  is  known 
to  be 

z=M^iV)^F{^-irj),  (17) 


$66  CALCULUS   OF  VARLATLONS. 

/and  F  denoting  any  functions  whatever,  real  or  imaginary.  See  De  Morgan's 
Diff.  and  Integ.  Calc,  pp.  723,  719,  putting  i  for  a^^and  rj  for  t.  See  also  Boole's 
Diff.  Eqs.,  Chap.  XV. 

It  is  evident  that,  having  differentiated  (14),  the  present  integral  is  more  gen- 
eral than  the  integral  of  that  equation  ;  but  it  includes  the  equation  of  the  re- 
quired surface,  which,  when  the  forms  of /and  i^are  assigned,  must  be  deduced 
in  the  following  manner.     We  have,  from  the  last  of  equations  (10), 


Z=  I  z  cos  17]  dr}. 


(18) 


in  which,  by  (17),  z  will  become  a  known  function  of  ^  and  r}.  Then  we  shall 
obtain  t,  by  integrating  with  respect  to  7}  only,  observing  to  add  to  the  result  an 
arbitrary  function  of  |,  which  function  must  be  then  so  determined  as  to  satisfy 
(14),  otherwise  the  value  of  C  will  be  too  general.  Now  substituting  this  final 
value  of  C  in  equations  (9)  and  (10),  and  then  eliminating  ^  and  7],  we  shall  obtain 
the  particular  equation  sought. 
As  an  example,  assume 


Whence,  by  (17)  and  (18), 

^  =   I  {a^-\-  brf)  cos  ir}  drj  =  —  {ac,  -}-  brj)  i  sin  it}  —  b  cos  ir]  -f-  X, 

X  being  any  function  of  I  ;  its  first  and  second  differential  coefficients  with  re- 
spect to  I  being  X'  and  X" .  Now  using  this  value  of  C  in  equations  (11),  we  find 
easily 

u:=^  —  bzas^iri -\- X" -\- X,         w  =.b  cos  ir}, 

and 

ti-\-w  =  X"-\-X=o. 

The  integral  of  this  equation  is,  as  the  reader  can  easily  verify  by  dififerentiation, 
X  =z  c  cos  ^-|-^' sin  ^,  <r  and  ^' being  any  arbitrary  constants.  If  now  we  replace 
X  by  this  value,  we  shall  have  the  true  expression  for  C  belonging  to  the  parti- 
cular surface  sought.  For  greater  simplicity,  let  us  now  suppose  c  and  c'  to  be- 
come zero,  so  that  X  will  vanish  also,  and  the  value  of  C  will  become 

C  =  —  {aq-\-  bij)  i  sin  irj  —  b  cos  i?/  =  —  zi  sin  ir/  —  b  cos  ir/.  (ig) 


NOTES.  567 

Now  resuming  equation  (9)  and  the  first  equation  (10),  and  eliminating  between 
them  X  and  y  alternately  by  multiplication,  and  substituting  for  C  from  (19),  and  for 

— -  its  value  —  ai  sin  it],  we  shall  obtain 


X  —  b  cos  IT]  cos  I  —  ai  sin  ii]  sin  |, 
y  —  b  cos  irj  sin  ?  -|"  ^^'  ^in  ^'^  cos  I, 
z  =  aq-}-  brj. 


(20) 


the  last  equajtion  having  been  obtained  before.  Or  if  in  the  same  equations  we 
write 

X  =-rQOS,(a),         y  T=  rsmoo, 
observing  that 

xcos^-{-ysxn  |  =  r(cos  oj  cosl4-  sin  go  sin  ?)  =  rcos(G!?  — ^), 

X  sin  ^  —  y  cos  S  =  r  (cos  go  sin  |  —  sin  go  cos  ^)  =  r  sin  {co  —  ^), 

we  have  in  polar  co-ordinates,  z  remaining  as  before, 

rcos(o!3  —  ^)  =  ^cos/77,         r  sin  {go  —  ^)^^  ai  sin  irj,         z-=  a^-\-  brj.       (21) 

If  now,  in  equations  (20)  or  (21),  we  can  eliminate  q  and  rj,  we  shall  obtain  a  par- 
ticular equation  of  the  minimum  surface.  Suppose  we  make  «  zero.  Then  (21) 
gives 

GO  —^  =  0        and     V  =  -i^> 
and  by  the  first  of  equations  (4)  we  shall  have 


r  =  b cos  —  =:  -\^-\-e    f> 
b        2\ 

which  is  evidently  the  equation  of  the  surface  generated  by  the  revolution  of  a 
catenary  about  the  axis  of  z,  that  axis  coinciding  with  the  directrix  ;  which  is  the 
same  result  as  has  been  previously  obtained.  Making  b  zero  while  a  is  not,  we 
have 

^  It  .  Z  TC     .     Z 

G3—  |=— ,  1  =  -,  C£>=  — +  -, 

2  a  2.        a 

the  equation  of  a  helicoid. 

When  a  and  b  are  any  constants  whatever,  we  can  still  eliminate  ^  and  r]  from 
(21).  As  the  result  merely  is  given  by  Moigno,  we  will  here  indicate  the  work 
without  explanation.     We  have 


568  CALCULUS   OF  VARIATIONS, 

r^  =  b^  cos^  27]  —  a^  sin^  irf  =  r^  (sin^  it]  -\-  cos^  irf), 
r^  +  a2  =  (^2  4.  ^2).cos2  irj^         r''-b^z=-  («2  _|_  ^2)  gin^  iy. 


i  tan  z7?  =  i/--__i!  tan  {Go-^)  =  \i  tan  ?7, 


<j!^  —  aci)  =  —  a  tan ' 


(?»tan;^)=-«tan    '-(e/^ZJj, 


/sin/77  =  -  (<?-^  —  <?»?),        cos?7  =  -  (^->7  +  ^)»        COS /t;  4"  2  sin  ?77  =  (f — n, 


^77=  —  ol.  (cos  ev  +  e  sin  im  =  —  5/ ■ — =z  —  a^. 


Whence  we  obtain 

z  —  aoa  =  — «tan— * 


]by  ^+a^\  4/^qr^  ^^^^ 


This  equation  evidently  represents  a  surface  generated  by  a  helicoid  move- 
ment about  the  axis  of  z,  of  a  plane  curve  whose  equation  will  be  given  by  (21)  if 
in  it  we  make  go  zero,  and  r  equal  to  x. 


NOTE  TO  ART.  372. 

When,  however,  m  =  —  2,  equation    (7)  will   not  give  the  true   solution.      But 
by  integrating  for  this  case  separately,  just  as  before,  we  shall  obtain  2J3 

'  <?•'■<  !.-^   OF  THR 


rlFOHit 


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